The conjugacy problem for Coxeter groups - Warwick Insitemasbal/index_files/me.pdf · 2008-06-18 · The conjugacy problem for Coxeter groups Daan Krammer June 18, 2008 University

The conjugacy problem for Coxeter groups

Daan Krammer

June 18, 2008

University of WarwickMathematics DepartmentCoventry CV4 7ALUnited [email protected]

Abstract

This is largely my 1994 thesis supervised by Arjeh Cohen. Weprove that for every finite rank Coxeter group there exists a polyno-mial (cubic) solution to the conjugacy problem.

Contents

1 Coxeter groups 121.1 Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Root bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Convexity and gatedness . . . . . . . . . . . . . . . . . . . . 151.4 Reflection subgroups . . . . . . . . . . . . . . . . . . . . . . 171.5 Nonnegative matrices . . . . . . . . . . . . . . . . . . . . . 20

2 On the Tits cone 202.1 The Galois correspondence . . . . . . . . . . . . . . . . . . 202.2 The interior of the Tits cone . . . . . . . . . . . . . . . . . . 22

3 The normalizer of a parabolic subgroup 243.1 The normalizer of a parabolic subgroup . . . . . . . . . . . 243.2 Finite subgroups of Coxeter groups . . . . . . . . . . . . . . 28

4 Hyperbolic reflection groups 294.1 Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Affine Coxeter groups . . . . . . . . . . . . . . . . . . . . . 294.3 Hyperbolicity and word hyperbolicity . . . . . . . . . . . . 304.4 Removed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.5 Hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . 314.6 Hyperbolic reflection groups . . . . . . . . . . . . . . . . . . 324.7 The conjugacy problem . . . . . . . . . . . . . . . . . . . . 33

2 Daan Krammer

5 The axis 355.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Periodic, even and odd roots . . . . . . . . . . . . . . . . . 355.3 Distinguishing even roots from odd ones . . . . . . . . . . . 375.4 A pseudometric on U0 . . . . . . . . . . . . . . . . . . . . . 385.5 The axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6 Non-emptiness of the axis . . . . . . . . . . . . . . . . . . . 455.7 Critical roots . . . . . . . . . . . . . . . . . . . . . . . . . . 475.8 The parabolic closure . . . . . . . . . . . . . . . . . . . . . 515.9 Calculations in number fields . . . . . . . . . . . . . . . . . 535.10 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6 The conjugacy problem 566.1 Dividing out the radical . . . . . . . . . . . . . . . . . . . . 566.2 Ordered triples of roots . . . . . . . . . . . . . . . . . . . . 576.3 Outward roots and their ordering . . . . . . . . . . . . . . . 596.4 First solution to the conjugacy problem . . . . . . . . . . . 636.5 The greatest eigenvalue . . . . . . . . . . . . . . . . . . . . 666.6 Removed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.7 Second solution to the conjugacy problem . . . . . . . . . . 726.8 Free abelian groups in Coxeter groups . . . . . . . . . . . . 74

A Euclidean complexes of non-positive curvature 76A.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 76A.2 Metric complexes . . . . . . . . . . . . . . . . . . . . . . . . 76A.3 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78A.4 Convex metrics . . . . . . . . . . . . . . . . . . . . . . . . . 79

B The Davis-Moussong complex 79B.1 Nerves of almost negative matrices . . . . . . . . . . . . . . 79B.2 Cells for the Davis-Moussong complex . . . . . . . . . . . . 80B.3 The Davis-Moussong complex . . . . . . . . . . . . . . . . . 82B.4 The embedding of the Davis-Moussong complex in the Tits

cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83B.5 Comparison of dW and dM . . . . . . . . . . . . . . . . . . 84B.6 The conjugacy problem . . . . . . . . . . . . . . . . . . . . 87

References 87

Index 91

List of symbols 93

The conjugacy problem for Coxeter groups 3

Dependence of sections and appendices.

1

23

4 5

6

A

B

Notational conventions. More specific notational conventions canbe found on page 93.

#X cardinality of a set X〈X〉 group or root system generated by a set XwZ group generated by an element w\ division by a left group action− set difference⊂ (not necessarily strict) inclusion⊔ disjoint unionX0 topological interior of X

X topological closure of XC(w) conjugacy classN(H) normalizerZ(H) centralizerG⋉N semi-direct product with normal subgroup NAnn annihilator for dual vector spacesCh convex hullIm image of a mapO(k), O(k, ℓ) real orthogonal groupsP (X) power set of XSpan linear spanSym(X) symmetric group on X

4 Daan Krammer

Introduction

Algorithms. There exists a precise definition of what an algorithmis ([40, Chapter 5]), but for reading this paper it is enough to have as littlefamiliarity with the concept as one acquires after writing a few computerprograms.

More important to us than whether an algorithm exists is whether afast algorithm exists: the complexity. An algorithm is called polynomial ofexponent k if t(n) = O(nk) where t(n) is the maximal time taken by thealgorithm on input of length n > 0. An algorithm is called exponential ifwe have t(n) = O(An) for some A > 1.

These are upper bounds for t(n). If we are interested in lower bounds,we say so explicitly.

Algorithmic problems for groups. In 1912, Max Dehn [26] stated,and in certain cases solved, three fundamental problems on groups, two ofwhich are as follows. Let G = 〈X |R 〉 be a (say finitely) presented group.Thus, X is a finite set, and R is a finite subset of the free group F (X) onX, and G = F (X)/N , where N is the smallest normal subgroup of F (X)containing R. Let π: F (X) → G denote the natural map.

Word problem : Does there exist an algorithm that, given w ∈F (X), decides whether or not πw = 1?

Conjugacy problem : Does there exist an algorithm that, givenv, w ∈ F (X), decides whether or not πv and πw are conjugatein G?

If an algorithm for the word problem exists, we say that the group hassolvable word problem, and similarly for the conjugacy problem.

Note that if the conjugacy problem for a certain group is solvable, thenso is the word problem. It is easy to see that the solvability of the twoproblems does not depend on the given presentation, only on the group G.For a survey on the conjugacy problem, we refer to [37].

In order to make the complexity of the Dehn problems well-defined, weneed to define what their input lengths are, at least up to reparametrisationby a polynomial.

Let ℓ: F (X) → Z be the length with respect to X. We saw that theinput of the word problem is an element g ∈ F (X); the length of this inputis defined to be ℓ(g). Likewise, the input of the conjugacy problem is a pair(g, h) ∈ F (X) × F (X) and we define its length to be ℓ(g) + ℓ(h).

This is a common choice but by no means the only possible one.

The word problem for Coxeter groups. A Coxeter matrix is asymmetric matrix M = (mst)s,t∈S with mst ∈ {1, 2, . . .}∪{∞} and mst =1 ⇔ s = t. To a Coxeter matrix M we associate a Coxeter group given bythe presentation

W = 〈S | {(st)mst | s, t ∈ S} 〉.Our main references for Coxeter groups are [8], [19], [36], [47]. See also

[7], [25], [33], [48].Two solutions to the word problem for Coxeter groups were found by

Tits. One of them is a consequence of his result that W is isomorphic


to a subgroup of GL(n,Q) (corollary 1.2.3) where Q denotes an algebraicclosure of Q. It is easy to deduce a polynomial (indeed, quadratic) solutionto the word problem.

Another solution [46] is elegant because it uses only manipulations ofwords in X. But it is slow: it has an exponential lower bound for time.

In 1993, Brigitte Brink and Bob Howlett, building on ideas of MikeDavis and Mike Shapiro, proved that finite rank Coxeter groups are auto-matic [10]. Every automatic group has a quadratic solution to the wordproblem and the involved algorithm is clean and easy to implement [32],[34].

The conjugacy problem for Coxeter groups. In his 1988 the-sis [43] supervised by Mike Davis, Gabor Moussong proved that every fi-nite rank Coxeter group acts properly cocompactly on a CAT(0) Euclideancomplex. (An action G×X → X is called proper if for every compact setK ⊂ X, the space {g ∈ G | K ∩ gK 6= ∅} is compact.) It is easy to deducea solution to the conjugacy problem for Coxeter groups. Our version ofthese ideas can be found in appendix B. Also see the 2007 monograph [25]by Mike Davis.

This solution to the conjugacy problem was the only one so far, and ithas both upper and lower time bounds which are exponential.

One of our main aims is to give a polynomial solution (theorem 6.7.4).It has a lot in common with Tits’s first solution to the word problem: itmakes extensive use of Tits’ faithful representation of Coxeter groups; it ispolynomial in time; and it involves algorithms in finite field extensions ofQ, making it more wieldy to implement than one might have hoped.

The bulk of the paper doesn’t mention words like algorithm and com-plexity though and is devoted to proving theorems as usual.

One of our main results is theorem 6.5.14 which states the following.Let W → GL(E∗) be the Tits representation of a Coxeter group W (seesection 1 for details). Suppose that W is irreducible and not affine. Letw ∈ W not be contained in a parabolic subgroup of W other than W (seesection 2). Then there is a 2-dimensional w-invariant subspace Ew ⊂ E∗

meeting U0, the interior of the Tits cone U .

The algebraic rank of a group G is the supremum of those n for whichZn can be embedded in G. Another main result is theorem 6.8.3. It im-plies that one can read off the algebraic rank of a Coxeter group from theCoxeter matrix. It explicitly gives the commensurability classes of abeliansubgroups (two subgroups of W are called commensurable if their intersec-tion has finite index in either).

What we don’t do. Our result leaves the following three questionsopen.

(a) We prove that every fixed Coxeter group has a polynomial wordproblem. In other words, the bounds depend on the group. What ifthe Coxeter group is allowed to vary?

(b) Find an algorithm which is easier to implement than ours.Is there a property that groups may or may not have (call it

property P) such that for every group with property P there exists

6 Daan Krammer

a polynomial solution to the conjugacy problem which is easy to im-plement? If so, can you prove that Coxeter groups have property P?

As to the word problem, one may choose property P to be auto-maticity. Coxeter groups are known to be automatic [10].

(c) One may define the input length of a solution to a Dehn problemdifferently, resulting in a non-equivalent notion of complexity. In-stead of the word metric with respect to a finite set X of generators,we suggest the straight line length with respect to X: the straightline length of an element g ∈ F (X) is the least k > 0 for which thereexist a1, . . . , ak ∈ F (X) such that ak = g, and for all p ∈ {1, . . . , k},either ap ∈ X or there are i, j < p such that ap = ai aj or ap = a−1

i .

Comparison with the thesis. This paper is essentially my 1994thesis. For ease of reference, I have kept the numbering of the originalthesis as far as possible, resulting in some empty subsections. The maindifferences with my thesis are as follows.

I rewrote the introduction.

We removed sections 4.4, 6.6 and 6.9–6.11.

Appendix A has been replaced with a mere summary on CAT(0) Eu-clidean complexes, referring to the literature for all the proofs.

We slighly changed the statement of 6.8.2 and the proof of 6.8.3 to fixsome unclarity in the proof of 6.8.3.

Many thanks to Bernhard Muhlherr who suggested to include 6.1.2.This also cancels the original item by that number, and simplifies the proofof 6.1.3.

Acknowledgement. As noted above, this paper is essentially my1994 PhD thesis supervised by Arjeh Cohen at Utrecht. The paper owes anenormous lot to his insights. I am deeply indebted to Arjeh for his carefulguidance. It was a wonderful time working with him.

Detailed overview by section

Section 1. This section collects the preliminaries with references tothe literature for their proofs.

For the purpose of this introduction, we mention the following.

A central notion is a root basis, defined in 1.2.1. Every root basis givesrise to a faithful representation of an associated (finite rank) Coxeter groupW in a finite dimensional real vector space E∗.

The Tits cone is a certain convex cone U ⊂ E∗ left invariant by W .The W -action on the interior U0 is proper.

A reflection in W is a conjugate to an element of S. Every reflectionin W gives rise to a hyperplane: the set of fixed points in U . It hascodimension 1.

The set of such hyperplanes is locally finite in U0. A chamber is aconnected component of U minus all hyperplanes. The Tits cone is theunion of the closed chambers.

Section 2. In section 2 we prove some results on the Tits cone whichhave been known but deserve to be looked at again.


A root basis (E, (·, ·),Π) is said to be non-degenerate if (·, ·) is. Weprove the (crucial) results of subsection 6.3 under the assumption thatthe root basis is non-degenerate. Luckily, as we prove in section 6.1, forevery irreducible non-affine Coxeter group there exists a non-degenerateroot basis. But it may not be a classical root basis, that is, a root basiswhere Π is a basis for E. In section 2 we pay special attention to non-classical root bases.

For a subset X ⊂ U we define X ′ ⊂W to be the pointwise stabilizer inW of X. Conversely, for a subset H ⊂ W we define H ′ ⊂ U to be the setof points fixed by each element of H. Sets of the form X ′ or H ′ are calledstable. The prime maps set up a bijection between stable subsets of U andstable subsets of W .

In the case of a classical root basis, the stable subsets of W are preciselythe parabolic subgroups of W . It follows that every subset X ⊂ W iscontained in a unique smallest parabolic subgroup: its parabolic closurePc(X). The parabolic closure Pc(w) of a single element w ∈ W is at thecenter of our attention in later sections.

For non-classical root bases, a parabolic subgroup may not be stable;see 2.1.4 for the correct statement in this direction.

Section 3. We recall in 3.1.3 an important theorem by Deodhar [27],earlier proved by Howlett [35] for finite Coxeter groups. Consider thegroupoid (=category whose morphisms are invertible) whose objects arethe subsets of S and such that the set of morphisms from I to J is the setof those g ∈ W for which gWI g

−1 = WJ . The result gives an explicit setof generators of this groupoid. In particular, it tells us whether WI , WJ

are conjugate in W .

Using this result, after a little work we arrive at corollary 3.1.13 whichstates the following. The conjugacy problem for Coxeter groups can besolved as soon as for all Coxeter groups, the following two problems can.

(1) Given w ∈W , determine I ⊂ S, g ∈W such that Pc(w) = gWIg−1.

(2) Given v, w ∈W such that Pc(v) = Pc(w) = W , decide whether v, ware conjugate in W .

A polynomial algorithm for (1) is obtained later on in 5.10.9 and for (2) in6.4.4 as well as in 6.7.4.

In subsection 3.2 we study finite parabolic subgroups. In particular, wesolve both (1) and (2) if v, w are torsion (3.2.1(b) and 3.2.2).

Section 4. By a hyperbolic reflection group we mean a discrete sub-group of the isometry group of hyperbolic space, generated by finitely manyreflections. Such a group is always a Coxeter group.

This should not be confused with word hyperbolic groups. See subsec-tion 4.3 for a definition of this and a comparison of the two concepts.

Suppose that (E, (·, ·),Π) is a root base for W such that the symmetricbilinear form (·, ·) has signature (n, 1). Then W is a hyperbolic reflectiongroup; indeed hyperbolic space Hn is one of the two components of {R>0x |x ∈ E, (x, x) < 0}.

It is known that if g is an isometry of hyperbolic space Hn then preciselyone of the following is true (see 4.5.1).

(a) g has a fixed point in Hn.

8 Daan Krammer

(b) g has no fixed points in Hn, and has exactly one fixed point atinfinity.

(c) g has no fixed points in Hn, and has exactly two fixed points atinfinity.

In the above cases we call g, respectively, elliptic, parabolic, and hyperbolic.We find this trichotomy a useful guidance. In section 6 we will find a

rather similar trichotomy for Coxeter groups.The main result of section 4 is 4.7.3 stating that if w ∈W is parabolic in

the above sense and fixes R>0x ∈ Hn, then x ∈ U . In theorem 4.7.6 we usethis to give a solution to the conjugacy problem for hyperbolic reflectiongroups.

Section 5. For a root α, let fα: U0 → {−1/2, 0, 1/2} be the map

fα(x) =

−1/2 if 〈x, α〉 < 0,0 if 〈x, α〉 = 0,1/2 if 〈x, α〉 > 0.

We define pseudometrics dα and dΦ on U0 by

dα(x, y) = |fα(x) − fα(y)|, dΦ(x, y) =∑

α∈Φ+

dα(x, y).

Then dΦ(x, y) = 0 if and only if {x, y} is contained in some facet. Moreover,dΦ generalizes the metric dW in the sense that dW (x, y) = dΦ(xC, yC) forall x, y ∈W .

In this section, we fix an element w ∈W . We define the axis Q(w) by

{x ∈ U0 | for all n ∈ Z: dΦ(x,wnx) = |n| dΦ(x,wx)}.

Examples of pictures of Q(w) can be found in figures 5–7. We prove thatQ(w) is non-empty in 5.6.10.

A large part of section 5 is devoted to distinguishing different sorts ofroots with respect to w. For example, a root α is said to be w-periodic ifthere exists n > 0 such that wnα = α.

We often say periodic instead of w-periodic if there is no danger ofconfusion, and likewise for the other classes of roots.

There is a bijection between the set Φ of roots and the set of half-spaces.If the root α corresponds to the half-space A then we say that A satisfiessome property (such as being periodic) if α does. Also, we have a 2-to-1map from Φ to the set of reflections in W . If α has a property then we saythat the corresponding reflection rα has, provided this doesn’t lead to anyconfusion.

In figure 1.2 we give an overview of the classes of roots and their defi-nitions.

In theorem 5.8.3 we prove that the parabolic closure Pc(w) of w is thesubgroup of W generated by the w-essential reflections. Also, Pc(w) is adirect product Pc∞(w) × Pc0(w) where Pc∞(w) is generated by the oddreflections and Pc0(w) by those reflections rα for which α is perpendicularto all odd roots.

In subsection 5.10 we prove that there are polynomial algorithms doingall sorts of related computations. For example, on input w,α one algorithm


Figure 1: Classes of roots and their definitions

root ��

BBBBBBB

even

periodic

odd

��

@@

��HH

��HH

α or −α supporting (and periodic)

rest

critical

supporting (and even)

non-supporting (and even)

outward

inward

essential

Let α be a root and let A denote the corresponding half-space.We say that A is periodic if there exists n > 0 such that wnA = A.We say that A is even if it is not periodic and for all x ∈W , the number

#{n ∈ Z | A separates wnx from wn+1x} is finite and even.We say that A is odd if it is not periodic and for all x ∈W , the number

#{n ∈ Z | A separates wnx from wn+1x} is finite and odd.We say that A is supporting if Q(w) ⊂ H(α) := {x ∈ U0 | 〈x, α〉 > 0}.We say that A is critical if it satisfies the following equivalent conditions.

• SpanwZα is positive definite and∑k

n=1wnα = 0, where k is the

smallest positive integer with wkα = α.• ⋂

n∈Z wnA =

⋂n∈Z w

n(W −A) = ∅, and A is periodic.• Q(w) ⊂ µ(α).

As the diagram suggests, a periodic root α is called rest if it is notcritical and neither α nor −α are supporting.

An odd root α is said to be outward if for some (hence all) x ∈ U0 thefollowing holds. For almost all n ∈ Z, we have n〈wnx, α〉 < 0. It is calledinward if −α is outward.

A root is called essential if it is odd or critical.

10 Daan Krammer

(5.10.1) decides to which of the classes the root α belongs. Another exampleis an algorithm (5.10.9) that, on input w, finds I ⊂ S and g ∈W such thatPc(w) = gWI g

−1. Most of the algorithms make extensive use of the Titscone.

Section 6. In 3.1.13 and 5.10.9, the conjugacy problem for Coxetergroups has been reduced to the case Pc(v) = Pc(w) = W . In section 6, wesolve the conjugacy problem for this case.

We treat affine and non-affine groups separately. The affine groups havebeen taken care of in 4.2.1.

In subsection 6.1, it is shown that for irreducible infinite non-affineCoxeter groups, there exists a non-degenerate root basis, which means thatthe radical E⊥ = {x ∈ E | (x, y) = 0 for all y ∈ E} is zero. Indeed, everyroot basis E gives rise to one, obtained by dividing out the radical.

In subsection 6.2 we prove some useful inequalities, such as corollary 6.2.3which states that if α < β < γ are roots then (α, β) ≤ (α, γ) and (β, γ) ≤(α, γ).

From here on, we assume (W,S) to be irreducible, infinite and non-affine, unless stated otherwise. Furthermore, an element w ∈ W withPc(w) = W is fixed. By 5.8.7, we have Pc∞(w) = W , and Span(Π) isspanned by the outward roots.

By 6.1.3 there exists a non-degenerate root basis (E, (·, ·),Π) for W .Also we may assume that E is spanned by Π and hence by the odd roots.Such E is fixed from now on.

A key subsection is 6.3. One useful result from this subsection is corol-lary 6.3.8 stating that for all outward α, β one has (α,wnβ) → ∞ as n→ ∞or n→ −∞. Another is corollary 6.3.11 which says that for all outward α,β one has w−nα < β < wnα for almost all n > 0. A result of independentinterest is corollary 6.3.10 stating that wZ has finite index in the centralizerof w.

In subsection 6.4 we prove that every Coxeter group of rank r has apolynomial time conjugacy problem with exponent r + 3 (theorem 6.4.4).This is not our fastest algorithm, which will be obtained in a later subsec-tion. Another result, unrelated to polynomial algorithms, is corollary 6.4.7stating that for every finite rank Coxeter group W there exists a constantN = N(W ) such that if v, w ∈ W are conjugate, then there exists g ∈ Wwith gvg−1 = w and ℓ(g) ≤ N(ℓ(v) + ℓ(w)).

We turn to subsection 6.5. Let λ denote the maximum of the absolutevalues of the complex eigenvalues of w. Let Eµ ⊂ E∗ denote the generalizedeigenspace of eigenvalue µ, for all µ.

A relatively easy lemma (6.5.4) states that λ > 1, that λ is an eigenvalueof w, and that if µ is any eigenvalue of w such that |µ| = λ, then (w −µ)Eµ = 0.

The larger part of subsection 6.5 is devoted to proving that λ is asimple eigenvalue. This is proved in theorem 6.5.14, which also yields thatthe set Q0(w) := (Eλ ⊕ Eλ−1) ∩ U0 is 2-dimensional, indeed, of the formR>0u1 ⊕ R>0u2 for independent u1, u2 ∈ E∗. It is the closest possibleanalogue to the axis in real hyperbolic space of a hyperbolic element. It isclear that Q0(w) ⊂ Q(w).

Theorem 6.7.4 states that for every (finite rank) Coxeter group thereexists a cubic solution to the conjugacy problem, that is, a polynomial one


with exponent 3. The proof relies on our knowledge of Q0(w).

Subsection 6.8 presents some results of independent interest. They arenot related to algorithms, and are relatively easily proved using some ofour results.

Let I1, . . . , In ⊂ S be irreducible, non-spherical and pairwise perpendic-ular. For all i, let Hi ⊂ WIi

be a subgroup as follows. If Ii is affine, Hi isthe translation subgroup. Otherwise, Hi is any subgroup of WIi

isomorphicto Z. We call

∏iHi a standard free abelian subgroup.

The main result of this subsection is theorem 6.8.3 which states thefollowing. Each free abelian subgroup of a finite rank Coxeter group W hasa finite index subgroup which is conjugate to a subgroup of a standard freeabelian subgroup.

Appendix A. In this appendix we collect all definitions and resultsone needs to know to be able to read appendix B, and we point to theproofs in the literature. Our main reference is [9].

The appendix begins with a short introduction to metric spaces andtheir geodesics in subsection A.1. A geodesic space is a metric space inwhich any two points can be connected by a geodesic.

A Euclidean cell is the convex hull of finitely many points in Euclideanspace En with the usual metric. A Euclidean complex is what you get ifyou glue a bunch of Euclidean cells together by isometries of faces. Bya metric complex we mean a Euclidean complex or its cousin, a sphericalcomplex. In subsection A.2 we look at metric complexes.

A geodesic space X is called CAT(0) if, roughly, triangles in X areno fatter than in the Euclidean plane. For example, a simply connectedcomplete Riemannian manifold is CAT(0) if and only if it has nonpositivecurvature.

Just as in the tangent space to a point of a Riemannian manifolds thereis a unit sphere, every point in a spherical complex has a tangent-space-like-thing with a spherical complex in it. The latter is called the link, andit is a spherical complex. One can repeat the link operation to obtain morecomplexes of ever smaller dimensions; they are also called links.

As we are interested in constructing an action of a Coxeter group on aCAT(0) Euclidean complex, we need a way to prove that a Euclidean com-plex is CAT(0) if it is. This is provided by theorem A.3.2. The condition isthat every link satisfies the girth condition, that is, its minimal cycles havelength at least 2π.

Appendix B. In his 1988 thesis [43] supervised by Michael Davis,Gabor Moussong proved that every finite rank Coxeter group acts prop-erly cocompactly on a CAT(0) Euclidean complex. We call it the Davis-Moussong complex M . Appendix B presents our approach to it.

The appendix begins with subsection B.1 on certain spherical complexescalled nerves of almost negative matrices, or just nerves. Nerves satisfy thegirth condition, see B.1.1. Moreover, links of nerves are themselves nerves(B.1.2).

One difference between our approach and Moussong’s is that we buildM from one (Euclidean) cell for each finite parabolic subgroup, whereasMoussong builds his complex from many copies of one thing, one for each

12 Daan Krammer

element of W . The Euclidean cells involved in our construction are studiedin subsection B.2.

In subsection B.3 we construct the Davis-Moussong complex. Everylink in M is a nerve so M is CAT(0) (see B.3.3).

In subsection B.4 we construct an embedding f : M → U0 of the Davis-Moussong complex in the interior of the Tits cone. For every cell C ∈ M ,the bilinear form (·, ·) on Ann(C) ⊂ E is positive definite. It follows thatSpan(C) has a Euclidean metric. The metric on C inherited from Span(C)is the metric in M . This yields a different but equivalent construction ofthe Davis-Moussong complex.

One of the reasons we are interested in the Davis-Moussong complexis that it gives rise to a (slow) algorithm to the conjugacy problem forCoxeter groups. We prove this in proposition B.6.1 after some preparationin subsection B.5.

1 Coxeter groups

1.1 Coxeter groups

A Coxeter matrix is a symmetric matrix M = (mij)i,j∈X indexed bya set X, with entries in {1, 2, . . .} ∪ {∞} such that mij = 1 ⇔ i = j.The Coxeter group associated to M is the group generated by elementssi (i ∈ X) subject to the relations

(sisj)mij = 1.

If mij = ∞, there is no relation. The Coxeter group is denoted by W (M)or just W . Write S = {si | i ∈ X} ⊂ W . A Coxeter system is a pair(W ′, S′) of a group W ′ and a subset S′ ⊂ W ′ such that there exists aCoxeter matrix M and a group isomorphism W (M) → W ′, mapping S toS′. The cardinality of X is called the rank of the system (or group).

Let (W,S) be a Coxeter system. Following [8], we construct a represen-tation of W as follows. Let E be a real vector space with basis {ei | i ∈ X}.Define a symmetric bilinear form (·, ·) on E by (ei, ej) = − cos(π/mij). Welet si act on E as the orthogonal reflection in ei, that is, six = x−2 (ei, x)eifor all x ∈ E. It is easy to check that this extends to a representationW → GL(E).

In particular, the map i 7→ si is bijective. We prefer S over X as theindexing set, that is, we write mst = mij where s = si, t = sj , and es = eiwhere s = si. Another consequence of the representation is that mst is theexact order of st. Hence, the Coxeter matrix is determined by the Coxetersystem.

For w ∈ W , define the length ℓ(w) = min {k | w = s1 · · · sk for somes1, . . . , sk ∈ S}. We define a metric dW on W by dW (x, y) = ℓ(x−1y).

Thus, the metric space (W,dW ) is acted upon by W from the left bymultiplication. In the sequel, we will define two more metric spaces, actedupon by W . The actions will always be from the left.

The Cayley graph of (W,S) is the graph with vertex set W and edges{x, xs}, x ∈ W , s ∈ S. The edge {x, xs} is sometimes labelled s. Themetric dW is nothing but the path metric of the Cayley graph.

The Coxeter graph is the graph with vertex set S and labelled edges{{s, t} | mst > 2} with labels mst. Usually, labels 3 are omitted. The


Coxeter system (or group) is called irreducible if the Coxeter graph isconnected.

For a subset I ⊂ S, we denote the subgroup of W generated by I by WI .It is called a standard parabolic subgroup of W . Note that a subgroupof W is a standard parabolic subgroup if and only if it is connected in theCayley graph. A parabolic subgroup is a subgroup of W conjugate to astandard parabolic one.

1.2 Root bases

1.2.1 Definition. A root basis is a triple (E, (·, ·),Π) of a real vector spaceE, a symmetric bilinear form (·, ·) on E and a finite set Π ⊂ E such that

(1) For all α ∈ Π: (α, α) = 1.(2) For all different α, β ∈ Π:

(α, β) ∈ {− cos(π/m) | m ∈ Z>1} ∪ (−∞,−1].

(3) There exists x ∈ E∗ such that 〈x, α〉 > 0 for all α ∈ Π.Here, E∗ denotes the dual of E, and 〈·, ·〉: E∗ × E → R is the pairing.

We are grateful to Bernhard Muhlherr for pointing out that (3) is oftena consequence of (1) and (2). See 6.1.2 for the precise statement and aproof.

Probably, many of our results on root bases do not require the assump-tion that Π is finite. We will not go into this.

Note that Π does not have to span E, nor be independent. If (E, (·, ·),Π)is a root basis, then so is (Span Π, (·, ·)′,Π), where (·, ·)′ is the restrictionof (·, ·) to Span Π. An example of a root basis where Π is dependent isassociated to a quadrangle in the hyperbolic plane with angles π/mst (seesection 4.6). Another example will be given in 6.1.4.

Given a Coxeter matrix (mst)s,t∈S , a natural root basis is the onebriefly mentioned in the previous subsection, and which is defined by E =⊕

s∈S Res, (es, et) = − cos(π/mst) (= −1 if mst = ∞), Π = {es | s ∈ S}.Note that (3) is satisfied by x ∈ E∗ defined by 〈x, es〉 = 1 for all s ∈ S. Wewill sometimes refer to this as the classical root basis.

Fix a root basis (E, (·, ·),Π). For α ∈ Π, define rα ∈ GL(E) by rα(x) =x − 2(x, α)α. We define S = {rα | α ∈ Π}, W = 〈S〉 ⊂ GL(E). The dualaction of W ⊂ GL(E) on E∗ is defined by 〈wx,wy〉 = 〈x, y〉. Write es = αwhenever α ∈ Π, s = rα. For a subset I ⊂ S, write ΠI = {ei | i ∈ I}.We sometimes write ast = (es, et). For a subset I ⊂ S, define the Grammatrix AI = (ast)s,t∈I , and write EI = Span ΠI .

In the case where Π is a basis of E, it is sometimes convenient to usethe basis {fs} of E∗ dual to {es}, which is defined by 〈fs, et〉 = δst.

We define

C = {x ∈ E∗ | 〈x, es〉 > 0 for all s ∈ S}.

Observe C 6= ∅ by (3). For every I ⊂ S, define the face

CI =

{x ∈ E∗

∣∣∣∣〈x, es〉 = 0 for all s ∈ I〈x, es〉 > 0 for all s ∈ S − I

}.

We call a set I ⊂ S facial if CI 6= ∅. Note C∅ = C and C = ⊔I⊂SCI ,where a bar denotes topological closure. We define the Tits cone U =

14 Daan Krammer

⋃{wC | w ∈ W}. Its topological interior relative to E∗ is denoted by U0.A facet is a set of the form wCI , w ∈W , I ⊂ S. The dimension of a facetis by definition the dimension of its linear span. e For two points x, y in areal vector space, write [x, y] = {tx+ (1− t)y | t ∈ [0, 1]}. A subset X of areal vector space is called convex if for all x, y ∈ X, we have [x, y] ⊂ X.

1.2.2 Theorem. Let (E, (·, ·),Π) be a root basis, and retain the notationsof above. Then the following hold.

(a) (W,S) is a Coxeter system, and W is a discrete subgroup of GL(E).(b) Let w ∈W , s ∈ S. Then either

〈wC, es〉 ⊂ R>0 and ℓ(sw) > ℓ(w)

or

〈wC, es〉 ⊂ R<0 and ℓ(sw) < ℓ(w).

(c) Let w ∈W , I, J ⊂ S. If CI ∩ wCJ 6= ∅ then I = J and w ∈WI .(d) U is convex. If x, y ∈ U then there are only finitely many facets

meeting [x, y].

Proof. For the classical root basis, proofs can be found in [36, 5.13], [8,Th. 1, Ch. 4.4]. For other root bases, see [47] or apply 6.1.3. �

1.2.3 Corollary.

(a) For all x ∈ CI , the stabilizer in W of x is WI .(b) Let I ⊂ S. Then (WI , I) is a Coxeter system, and its length function

is the restriction to WI of the length function on W .(c) Let I, J ⊂ S. Then WI ∩WJ = WI∩J .(d) The word problem for Coxeter groups is solvable. �

By 1.2.3(c), if X ⊂ W is a subset, then there exists a smallest stan-dard parabolic subgroup of W containing X. It is called the standardparabolic closure of X.

1.2.4 Proposition. There exists an algorithm that, given x ∈ U , findsw ∈W such that wx ∈ C.

Proof. We describe the algorithm. Let x0 ∈ U be given, and put w0 =1 ∈ W . Compute xk, wk (k = 1, 2, . . .) which are (non-uniquely) definedas follows. Suppose xk 6∈ C. Choose s ∈ S such that 〈xk, es〉 < 0. Letxk+1 = sxk, wk+1 = swk. If xk ∈ C, then the algorithm finishes. Wehave xk = wkx0 for all k. We will show that the algorithm terminates.Let v ∈ W such that x0 ∈ vC. Choose y0 ∈ vC and write yk = wky0.Then for all s ∈ S, the condition 〈xk, es〉 < 0 implies 〈yk, es〉 < 0, whichshows that the sequence yk is also a sequence satisfying the conditions ofthe algorithm. Hence ℓ(wk) ≤ ℓ(v) whenever wk is defined. By 1.2.2(b),we have ℓ(wk) = k, which shows that the algorithm terminates. �

A root is an element in E of the form wes, w ∈ W , s ∈ S. The set ofroots is denoted by Φ and is called the root system associated to (W,S).To each α ∈ Φ we associate a half-space A(α) ∈W by

A(α) = {w ∈W | 〈wC,α〉 ⊂ R>0},


and a reflection rα ∈W by

rα(x) = x− 2(x, α)α.

NoteW = A(α)⊔A(−α) for every root α. Define Φ+ = {α ∈ Φ | 1 ∈ A(α)},Φ− = −Φ+, so that Φ = Φ+ ⊔ Φ−. Elements of Φ+ are called positiveroots, the others negative. If a root α =

∑s∈S xses is positive and Π is

independent, then xs ≥ 0 for all s ∈ S.A standard parabolic root subsystem is a set of the form ΦI =

{wes | w ∈W, s ∈ I}, where I ⊂ S. A parabolic root system is a set ofthe form wΦI , w ∈W , I ⊂ S.

The set of half-spaces is in bijection with Φ by α 7→ A(α). We sometimeswrite rA = rα if A = A(α). By 1.2.2(b), half-spaces can be characterizedcompletely combinatorially — for example, A(es) = {w ∈ W | ℓ(sw) >ℓ(w)}.

Let α ∈ Φ, A = A(α), x, y ∈W . We say that α (or A, or rα) separatesx from y if A contains precisely one of x, y. The distance dW (x, y) equalsthe number of reflections separating x from y.

1.2.5 Lemma. Φ is discrete in E.

Proof. Consider the map R: {α ∈ E | (α, α) = 1} → GL(E), α 7→ rα =(x 7→ x − 2(x, α)α). Since R is continuous and W ⊂ GL(E) is discrete,R−1(W ) is discrete in E and hence so is Φ ⊂ R−1(W ). �

1.2.6 Proposition. Let (W,S) be a Coxeter system and let E be theclassical root basis. The following conditions are equivalent.

(a) W is finite.(b) Φ is finite.(c) U = E∗.(d) C ∩ −U 6= ∅.(e) The bilinear form (·, ·) on E is positive definite.(f) There exists w ∈W such that for all s ∈ S: ℓ(ws) < ℓ(w).(g) There exists w ∈W such that for all v ∈W − {w}: ℓ(v) < ℓ(w).(h) There exists w ∈W such that wΠ = −Π.

Moreover, if these conditions hold, then the elements in (f), (g) and (h) areunique and equal.

Proof. A hint to a proof is given in [8, Ch. 5, p. 130, exercise 2]. �

The irreducible finite Coxeter groups have been classified by Coxeter[23].

We call I ⊂ S spherical if WI is finite. For every spherical I ⊂ S,there is a unique longest element in WI , by 1.2.6, which is denoted bywI .

In this thesis, unless stated otherwise, we fix a finite rank Coxeter sys-tem (W,S) and a root basis (E, (·, ·),Π) associated to it.

1.3 Convexity and gatedness

The results of this subsection will be used in 2.2.5, 3.1.8 and B.4.1.

1.3.1 Proposition. Let X ⊂W be a set. The following are equivalent.

16 Daan Krammer

(1) Let x, z ∈ X, y ∈ W with dW (x, z) = dW (x, y) + dW (y, z). Theny ∈ X.

(2) X is the intersection of a family of half-spaces.

Proof. (1)⇒(2). Let p ∈W −X. Choose x ∈ X closest to p. Let s ∈ S besuch that y = xs is closer to p than x is. Let A be the half-space containingx but not y. We have p 6∈ A. The proof will be finished once we haveshown X ⊂ A. Let z ∈ X − A. Note d(x, z) = d(x, y) + d(y, z). By (1),y ∈ X, a contradiction. Hence X ⊂ A, which concludes the proof. (2)⇒(1)is trivial. �

1.3.2 Definition. A set X ⊂W is called convex if it satisfies the propertiesof 1.3.1. Some authors call it geodesically closed. We call X gated if forevery x ∈ W , there exists y ∈ X such that for every half-space A withy ∈ A, x 6∈ A, we have X ⊂ A.

1.3.3 Proposition.

(a) A set X ⊂W is gated if and only if for every x ∈W , there exists y ∈X such that for all z ∈ X, we have dW (x, z) = dW (x, y) + dW (y, z).The points y of the two definitions are equal and unique.

(b) Gatedness implies convexity.

Proof. Left to the reader. �

1.3.4 Definition. Let I, J ⊂ S. We define DIJ to be the set of elements inW such that ℓ(sw) > ℓ(w) for all s ∈ I and ℓ(ws) > ℓ(w) for all s ∈ J .

1.3.5 Proposition.

(a) Let I, J ⊂ S, w ∈ W . Then the set WIwWJ ∩ DIJ consists ofprecisely one element. Every element of w ∈ W has a (not neces-sarily unique) expression w = udv, with u ∈ WI , d ∈ DIJ , v ∈ WJ ,ℓ(w) = ℓ(u) + ℓ(d) + ℓ(v).

(b) Let I ⊂ S. The map WI × DI∅ → W , (x, y) 7→ xy is a bijection.For all x ∈WI , y ∈ DI∅, we have ℓ(xy) = ℓ(x) + ℓ(y).

(c) WI is gated.(d) If w ∈ DIJ then WI ∩ wWJw

−1 = 〈I ∩ wJw−1〉.

Proof. For (a) and (b), see [8, exercise 1.3 Ch. 4]. Part (c) is easily seento be equivalent to (b). For (d), see [45, Lemma 2]. �

1.3.6 Corollary. Let w = s1 · · · sn, si ∈ S, ℓ(w) = n. Then the standardparabolic closure of w equals WI , with I = {s1, . . . , sn}.

Proof. [19, 5.1(ii)]. �

1.3.7 Corollary. Let w ∈ W , I = {s ∈ S | ℓ(sw) < ℓ(w)}. Then I isspherical and ℓ(w) = ℓ(wIw) + ℓ(wI).

Proof. By 1.3.5(b), there exist (unique) x ∈ WI , y ∈ DI∅ such thatw = xy, ℓ(w) = ℓ(x) + ℓ(y). For all s ∈ I, we have ℓ(sx) < ℓ(x) because,again using 1.3.5(b),

ℓ(sx) + ℓ(y) = ℓ(sxy) = ℓ(sw) < ℓ(w) = ℓ(x) + ℓ(y).


By 1.2.6, I is spherical, and x = wI . This proves the corollary. �

1.4 Reflection subgroups

We will give two basic theorems (1.4.2, 1.4.3) on reflection subgroups, es-sentially shown by Deodhar [28] and independently Dyer [30]. We study inmore detail subgroups generated by two roots in 1.4.7. This provides a linkbetween inner products of roots and the combinatorics of Coxeter groups,which will be used, among others, in 5.3.2, 5.6.5, 6.2.2 and 6.5.6.

1.4.1 Theorem. Let W be a group generated by a set S of involutions,and suppose that W acts on a set W ′. Let there be given p ∈ W ′, and foreach s ∈ S a set As ⊂W ′ such that the following hold.

(1) {p} =⋂

s∈S As.(2) W ′ = As ⊔ sAs for all s ∈ S.(3) If w ∈ 〈s, t〉, s, t ∈ S, then w(As ∩ At) is contained in either As or

sAs, and in the latter case, we have ℓ(sw) < ℓ(w).

Then (W,S) is a Coxeter system and the W -action on W ′ is simply tran-sitive.

Proof. The proof is essentially the same as that of [8, th. 1, p. 93]. �

Let (W,S) be a Coxeter system. To an edge {x, xs} in the Cayley graphof (W,S), we associate a reflection xsx−1. It is the (unique) reflectionreversing the edge. Any reflection is associated to at least one edge.

1.4.2 Theorem. Let X ⊂ W be a set of reflections, such that for allx, y ∈ X, we have xyx ∈ X. Let H = 〈X〉. Let Γ be the Cayley graph ofW minus those edges reversed by an element of X. Let H ′ be the set ofconnected components of Γ. Let p ∈ H ′ be the component containing 1.Let T be the set of reflections reversing an edge from p to W − p. Thenthe following hold.

(a) (H,T ) is a Coxeter system.(b) H acts simply transitively on H ′.(c) X equals the set of reflections in H.

Proof. For each t ∈ T , define At ⊂ H ′ by At = {wp | w ∈ H, t separates1 from w}. We leave it to the reader to check that conditions (1),(2),(3)of 1.4.1 are verified. Application of 1.4.1 immediately shows (a) and (b).In order to prove (c), let r ∈ H be a reflection. Let {x, xs} be an edgereversed by r. Hence x and xs are in different components of Γ, by (b) andsince r 6= 1. This shows r ∈ X. �

In the above theorem, T is called the canonical set of generatorsof H.

By a reflection subgroup of W we mean a subgroup generated byreflections. A root subsystem of Φ is a subset X ⊂ Φ such that for allα, β ∈ X, we have rαβ ∈ X. By 1.4.2(c), there is a bijection betweenreflection subgroups H ⊂ W and root subsystems of Φ, defined by H 7→{α ∈ Φ | rα ∈ H}.

Let X ⊂ Φ be a root subsystem and let H be the correspondingsubgroup. Let Y ⊂ X denote a minimal non-empty subset such that

18 Daan Krammer

Y ⊥∩X = X−Y . Then Y is called a component of X and 〈{rα | α ∈ Y }〉is a component of H. For subsets I ⊂ S, a component of I is bydefinition a component in the Coxeter graph on I. Note that WI is a com-ponent of WJ if and only if I is a component of J . We sometimes writeI⊥ = {s ∈ S | mst = 2 for all t ∈ I}.

A root subbasis of (E, (·, ·),Π) is a triple (E, (·, ·), X) (or simply X)such that X ⊂ Φ+ and for all different α, β ∈ X, we have

(α, β) ∈ {− cos(π/m) | m ∈ Z>1} ∪ (−∞,−1].

Note that (E, (·, ·), X) is indeed a root basis.

We will often consider root bases (E, (·, ·), X) where E and (·, ·) arefixed but X varies. The objects defined in this section, like Φ, U , C arethen written Φ(X) etc., in order to stress their dependence on X.

1.4.3 Theorem. There exists a bijection between root subbases and rootsubsystems of Φ with finitely many canonical generators, defined by X 7→〈X〉. Moreover, {rα | α ∈ X} is nothing but the canonical set of generatorsof the reflection group that it generates. For every set of roots Y generating〈X〉, we have #Y ≤ #X.

Proof. For the first two statements, see [30, theorem 4.4]. The inequality#Y ≤ #X is proved in [10, Lemma 1.5]. �

1.4.4 Lemma. Let (E, (·, ·), X) be a root basis, and let (E, (·, ·), Y ) be aroot subbasis. Let U(X), U(Y ), respectively, denote their Tits cones. ThenU(X) ⊂ U(Y ).

Proof. Let x ∈ U(X). Let g ∈ W be such that x ∈ gC. Let W,H denotethe Coxeter groups associated to X,Y , respectively. Let p,H ′ be as in1.4.2. Since H acts transitively on H ′, we may assume, after translationover an element of H, that g ∈ p. Equivalently, for all α ∈ Y , we have〈gC, α〉 ⊂ R>0. Hence 〈x, α〉 ≥ 0. Since this is true for all α ∈ Y , we havex ∈ C(Y ) ⊂ U(Y ). �

1.4.5 Lemma. Let α, β be two roots with (α, β) = −(p + p−1)/2, p > 0.Then, after replacing α, β by −α,−β if necessary, we have

〈U,α+ pβ〉, 〈U, pα+ β〉 ⊂ R≥0.

Proof. LetX be the root subbasis of Φ generating the same root subsystemas {α, β} does. It is easy to see that there exists w ∈ 〈rα, rβ〉 such that{wα,wβ} = X or −X. Thus, we may assume {α, β} = X. Let u = α+pβ,v = pα + β, V = {x ∈ E∗ | 〈x, u〉, 〈x, v〉 ≥ 0〉. First, we show that rα andrβ fix V . We have

rαu = rα(α+ pβ) = −α+ p (β + (p+ p−1)α) = p (β + pα) = pv,

and since rα is an involution, rαv = u/p. This shows that rα fixes V , andsimilarly for rβ. We have C(X) ⊂ V , whence U(X) =

⋃{wC(X) | w ∈

〈rα, rβ〉} ⊂ V , which proves 〈U(X), u〉, 〈U(X), v〉 ⊂ R≥0. By 1.4.4, we haveU ⊂ U(X), which finishes the proof. �


1.4.6 Definition. We define a (partial) ordering ≤ on Φ by α ≤ β ⇔ A(α) ⊂A(β).

To each root α ∈ Φ we associate open and closed half-spaces in U0

by

H(α) = {x ∈ U0 | 〈x, α〉 > 0}, K(α) = {x ∈ U0 | 〈x, α〉 ≥ 0}.

1.4.7 Proposition. Let α, β be two roots. Then the following hold.

(a) If |(α, β)| < 1 then all of the four intersections A(±α) ∩ A(±β) arenon-empty.

(b) (α, β) ≥ 1 if and only if α ≤ β or β ≤ α.(c) If α ≤ β, say (by (b)) (α, β) = (p+p−1)/2, p ≥ 1, then for all x ∈ U ,

we have

〈x, α〉 ≤ p〈x, β〉, 〈x, α〉 ≤ p−1〈x, β〉.

(d) α ≤ β if and only if 〈U, β − α〉 ⊂ R≥0.(e) Let X ⊂ Φ be a root subbasis, and let ≤

Xdenote the ordering on

the associated root system Φ(X). Then for all α, β ∈ Φ(X), we haveα ≤ β if and only if α ≤

Xβ.

Proof. (a). Let |(α, β)| < 1. Then P := Span {α, β} is positive definite.Hence the group F generated by rα and rβ is finite, because it is a dis-crete subgroup of the compact group {g ∈ GL(E) | gP = P, (gx, gy) =(x, y) for all x, y ∈ E} ∼= O(2). In order to prove that all four intersectionsare non-empty, suppose the contrary, say α ≤ β. Note α < β because|(α, β)| < 1. Now rβα < rββ, whence

α < β = −rββ < −rβα = tα,

where t = rβrα. Hence α < tα < t2α < · · · . Thus the root systemgenerated by α, β contains infinitely many roots tnα, a contradiction. Thisproves (a).

(b). ⇒. By 1.4.5, the set 〈U,α−pβ〉 is contained in R≥0 or R≤0. In thefirst case, for all x ∈ H(β), we have 〈x, α〉 ≥ 〈x, pβ〉 > 0, whence x ∈ H(α),and hence β ≤ α. The other case is treated similarly. ⇐. By (a), we have|(α, β)| ≥ 1. If (α, β) ≤ −1, then, by applying ‘⇒’ to α and −β, we findα ≤ −β or −β ≤ α. It is easy to check that this contradicts the fact thatα ≤ β or β ≤ α. Hence (α, β) ≥ 1.

(c). Same argument as ⇒ in (b).

(d). ⇒. Let x ∈ U . Let p ≥ 1 be as in (c). Then, by (c), 〈x, pβ−α〉 ≥ 0and 〈x, β − pα〉 ≥ 0. Adding these inequalities gives (p− 1)〈x, β − α〉 ≥ 0.If p > 1, then the right hand side of (d) follows. For p = 1, it followsimmediately from (c). ⇐ is trivial.

(e). ⇒. By (b), we have (α, β) ≥ 1. By (b) again, we have α ≤Xβ or

β ≤Xα. Suppose not α ≤

Xβ. Then, by (d), we have 〈U(X), α−β〉 ⊂ R≥0.

Since U ⊂ U(X) by 1.4.4, we find 〈U,α − β〉 ⊂ R≥0. However, by (d),applied to α ≤ β, we also have 〈U,α − β〉 ⊂ R≤0. Since U contains theopen set C, it follows that α = β, whence α ≤

Xβ, a contradiction. Hence

α ≤Xβ. ⇐ follows immediately from (d) and the fact that U ⊂ U(X),

1.4.5. �

20 Daan Krammer

1.5 Nonnegative matrices

The results of this subsection will be used in various places in this thesis,namely in the proofs of 2.2.2, 6.1.1 and B.4.1. See also subsection 6.5.

A non-negative, respectively, positive matrix is a square matrix overR all of whose entries are non-negative, respectively, positive. Similarly forvectors. Let A = (ast)s,t∈S be a non-negative matrix. It is called reducibleif there exists a partition S = I ⊔J , (I, J 6= ∅), such that ast = 0 whenevers ∈ I, t ∈ J . It is called irreducible otherwise.

1.5.1 Theorem (Perron-Frobenius). LetA be an irreducible non-negativematrix. Then up to multiples, A has a unique non-negative eigenvector v,say with eigenvalue λ. The vector v is positive. Moreover, every (complex)eigenvalue µ satisfies |µ| ≤ λ, and is simple if |µ| = λ.

Proof. [6, 1.4, p. 27]. �

1.5.2 Lemma. Let A = (aij) be a positive definite real square symmetricmatrix with aii = 1 and aij ≤ 0 (i 6= j). Then the entries of A−1 arenon-negative.

Proof. For a proof that uses the Perron-Frobenius theorem, see [43, 9.1]. �

Let E be a finite-dimensional real vector space. A cone in E is a subsetclosed under addition and multiplication by positive scalars. The dimen-sion of a cone is by definition the dimension of its span (the dimension ofthe empty cone being −1) A cone is called pointed if it does not contain alinear subspace of positive dimension. It is called solid if its interior rela-tive to E is non-empty. A proper cone is a closed pointed solid cone. Thetheorem of Perron-Frobenius has been generalized to linear maps mappinga proper cone into itself — see [6].

2 On the Tits cone

This section shows some results on the Tits cone. Many of the results wereproved by Vinberg [47].

2.1 The Galois correspondence

Let A,B be sets, and let R ⊂ A×B be a relation. The Galois correspon-dence associated to R is the pair of maps from P (A) to P (B) (power sets)and vice versa, both denoted by a prime, defined as follows. For X ⊂ A,

X ′ = {y ∈ B | (x, y) ∈ R for all x ∈ X}

and, similarly, for Y ⊂ B, Y ′ = {x ∈ A | (x, y) ∈ R for all y ∈ Y }.Usually, we write x′ instead of {x}′. For a set X which is at the same

time a subset of A and B (for example, X = ∅), X ′ has two meanings. Itwill however be clear which one is meant.

We collect a few properties about Galois correspondences. We haveX ′′′ = X ′ for all X ⊂ A or X ⊂ B. Sets of the form X ′ are called stable.A set X is stable if and only if X ′′ = X. More generally, X ′′ is the smalleststable set containing X. We call X ′′ the stable closure of X. Intersections


of stable sets are again stable, for⋂X ′

i = (⋃Xi)

′. The prime maps set upa bijection between the stable subsets of A and B.

The case of our interest is the Galois correspondence associated to therelation R between a Coxeter group W and the Tits cone U , defined by

(w, x) ∈ R⇔ wx = x.

First of all, note that stable sets are non-empty. Namely, stable sets inW contain 1, whereas stable sets in U contain 0. For every point x ∈ U ,the group x′ is a parabolic subgroup of W . Namely, if x ∈ wCI thenx′ = wWIw

−1, by 1.2.2(c).Write µ(α) = {x ∈ U | 〈x, α〉 = 0}. We call µ(α) the wall associated

to α.

2.1.1 Lemma. Let x, y ∈ U . Then x′ ⊂ y′ if and only if for all roots α, ifx ∈ µ(α) then y ∈ µ(α).

Proof. ‘If’. By 1.2.2(c), x′ and y′ are generated by the reflections thatthey contain. We have

rα ∈ x′ ⇔ x ∈ µ(α) ⇒ y ∈ µ(α) ⇔ rα ∈ y.

Hence x′ ⊂ y′. ‘Only if’ is easy. �

The following lemma will often be useful.

2.1.2 Lemma. Let X ⊂ U be a non-empty cone. Then there exists x ∈ Xwith x′ = X ′.

Proof. Let x ∈ X be such that x′ is minimal (such an x exists because ify ∈ U then y′ is a parabolic subgroup, and decreasing chains of parabolicsubgroups are stable). Suppose not x′ = X ′. Let y ∈ X such that x′ 6⊂ y′,say (by 2.1.1) y 6∈ µ(α), x ∈ µ(α). By 1.2.2(d), there exists z ∈ [x, y] suchthat for every root β, if z ∈ µ(β) then x, y ∈ µ(β). It follows that z′ ⊂ x′

by 2.1.1. Moreover, z′ 6= x′, because rα ∈ x′ − z′. Since X is a cone, wehave z ∈ X. This contradicts minimality of x′. Hence x′ = X ′. �

2.1.3 Definition. For I ⊂ S, write KI = {x ∈ U | 〈x, es〉 = 0 for all s ∈ I}.By a facial subgroup of W we mean a subgroup conjugate to WI for somefacial I ⊂ S.

2.1.4 Proposition. A subset of W is stable if and only if it is a facialsubgroup. A subset of U is stable if and only if it is of the form wKI , withw ∈W , I ⊂ S facial. Finally, the bijection is given by (wWIw

−1)′ = wKI .

Proof. Let I ⊂ S be facial. From 1.2.2(c) it follows that (KI)′ = WI .

Clearly, (WI)′ = KI . Translating the results over w ∈W gives (wWIw

−1)′ =wKI and (wKI)

′ = wWIw−1. This proves the ‘if’ parts. Now we will

prove the ‘only if’ parts. Let X ⊂ U be stable. By 2.1.2, there ex-ists x ∈ X with x′ = X ′. Let wCI be the facet containing x. ThenX = X ′′ = x′′ = (wWIw

−1)′ = wKI . In the last but one equality, 1.2.2(c)is used. Hence every stable subset in U is of the form wKI (with I facial).Consequently, each stable set in W is of the form (wKI)

′ = wWIw−1 with

I facial. �

22 Daan Krammer

In the special case of the classical root basis, the stable subgroups areprecisely the parabolic subgroups. It follows that the intersection of twoparabolic subgroups is again parabolic. Thus, every subset X ⊂W is con-tained in a unique smallest parabolic subgroup, which is called the para-bolic closure of X and written Pc(X). For root bases with the propertythat Π is independent, we have Pc(X) = X ′′. However, we will also beconsidering root bases without this property.

A formula for the intersection of two parabolic subgroups is given by1.3.5(d). Conversely, the intersection of two stable subsets of U is againstable. I do not know of a result on this intersection. In terms of subgroupsof W , the (equivalent) question is the following.

2.1.5 Question. What is the parabolic closure of the union of two parabolicsubgroups?

2.1.6 Theorem. Suppose W is irreducible and E = Span(Π). Then

U ∩ −U =

{{0} if W is infinite,

E∗ otherwise.

Proof. Let T = U ∩ −U . Then T is a W -invariant linear subspace of E∗.Write T ′′ = wKI . Since T is W -invariant, so is T ′′, whence T ′′ = KI . Wewill show

s ∈ I, mst > 2 ⇒ t ∈ I. (1)

Suppose s ∈ I, mst > 2. Let x ∈ KI . Since s ∈ I, we have 〈x, es〉 = 0.Since tT ′′ = T ′′, we have

0 = 〈tx, es〉 = 〈x, tes〉 = 〈x, es − 2 astet〉 = −2 ast〈x, et〉.Now ast = − cos(π/mst) 6= 0, so 〈x, et〉 = 0. We conclude KI ⊂ Kt. By2.1.4 we have Wt = (Kt)

′ ⊂ (KI)′ = WI , whence t ∈ I, which proves (1).

From (1) and irreducibility of W it follows that I = S or I = ∅. In thefirst case, we have T = {0} and W is infinite by 1.2.6. If I = ∅, we willshow that W is finite and T = E∗, which will prove the theorem. By 2.1.2,there exists x ∈ T such that x′ = T ′ = T ′′′ = (K∅)′ = W∅ = {1}. Hence xis on no wall; we may suppose x ∈ C. Hence x ∈ C ∩ −U . By 1.2.6, W isfinite and U = E∗. �

Theorem 2.1.6 has been proved by different methods by Vinberg, [47,Lemma 15, p. 1112].

2.2 The interior of the Tits cone

2.2.1 Definition. Let I ⊂ S be spherical. Let E⊥I = {x ∈ E | for all y ∈

EI : (x, y) = 0}. Note E = EI ⊕ E⊥I . Dually, E∗ = YI ⊕ ZI where

YI = Ann(E⊥I ) ⊂ E∗, ZI = Ann(EI) ⊂ E∗. We define pI : E → EI ,

qI : E∗ → ZI to be the projections with respect to these decompositions.

Equivalently,

qIx =1

#WI

∑

w∈WI

wx.

Note that for all x ∈ E∗, y ∈ E, we have

〈x, y〉 = 〈x, pIy〉 + 〈qIx, y〉. (2)


2.2.2 Lemma. Suppose that Π is a basis of E. Let I ⊂ S be spherical.

(a) For all s ∈ I, t ∈ S − I, we have 〈fs, pIet〉 ≤ 0.(b) For all s ∈ I, t ∈ S − I, we have 〈qIfs, et〉 ≥ 0.

Proof. (a). Let us write pIet =∑

s∈I bses. Then the bs are defined by theequations (eu, pIet − et) = 0 (u ∈ I). Thus,

∑s∈I asubs = atu (u ∈ I), or,

in matrix form, AIb = c, where c = (atu)u∈I . On inverting, b = A−1I c. By

1.5.2, the entries of A−1I are non-negative. Since t 6∈ I, the entries of c are

non-positive. Hence for all s ∈ I we have 〈fs, pIet〉 = bs ≤ 0.

(b). This follows from (a) and (2). �

2.2.3 Theorem. Let I ⊂ S be spherical. Then qI(C) = CI .

Proof. Let D = ⊕s∈SRds be the root basis with (ds, dt) = (es, et), andlet π: D → E be the linear map mapping ds to es. Let π∗: E∗ → D∗ bethe dual map defined by 〈π∗x, y〉 = 〈x, πy〉. For I ⊂ S, let BI be what isusually written CI , that is,

BI = {x ∈ D∗ | 〈x, ds〉 = 0 (s ∈ I), 〈x, dt〉 > 0 (t ∈ S − I)}.

Let rI : D∗ → D∗ be the projection, rIx = (

∑w∈WI

wx)/#WI . Since π∗ isa W -equivariant map, we have π∗ ◦ qI = rI ◦ π∗. For all x ∈ E∗, s ∈ S, wehave 〈qIx, es〉 = 〈qIx, πds〉 = 〈π∗(qIx), ds〉 = 〈rI(π∗x), ds〉, which shows

qIx ∈ CI ⇔ rI(π∗x) ∈ BI . (3)

Furthermore, we have 〈x, es〉 = 〈π∗x, ds〉, which shows

x ∈ C ⇔ π∗x ∈ B := B∅. (4)

By 2.2.2(b), we have

rI(B) ⊂ BI . (5)

Hence

x ∈ C(4)⇐⇒ π∗x ∈ B

(5)=⇒ rI(π

∗x) ∈ BI(3)⇐⇒ qIx ∈ CI ,

whence qI(C) ⊂ CI . Since qIx = x for all x ∈ CI ⊂ C, we have qI(C) = CI .Since qI is linear, qI(C) ⊂ CI . Using the fact that qI(C) is convex and densein CI , we will now show qI(C) = CI . Let x ∈ CI . Let O1, . . . , Ok be opensets in CI such that for every (x1, . . . , xk) ∈ O1 × · · · × Ok, the point x isin the convex hull of x1, . . . , xk. Since qI(C) is dense in CI , there existsxi ∈ Oi ∩ qI(C) for all i. Since qI(C) is convex, we have x ∈ qI(C). Weconclude qI(C) = CI . �

2.2.4 Corollary. Every spherical subset of S is facial.

Proof. Let I ⊂ S be spherical. By 2.2.3, CI is non-empty. Hence I isfacial. �

For another proof of 2.2.4, see [47, Theorem 7, p. 1114].

We recall that U0 denotes the topological interior of U relative to E∗.Using the fact that U is convex, it is easy to see that so is U0. The followingcorollary was proved by Vinberg [47, Theorem 2(3)].

24 Daan Krammer

2.2.5 Corollary. The cone U0 equals the union of the facets with finitestabilizer.

Proof. ⊃. Let I ⊂ S be spherical. Recall that (2.2.1)

qIx =1

#WI

∑

w∈WI

wx.

Using 2.2.3, it follows that CI is contained in∑

w∈WIwC. This is a subset

of U , which is open in E∗, and hence contained in U0. Translation overw ∈W shows wCI ⊂ U0.

⊂. Let x ∈ CI ∩ U0. Let B ⊂ U0 be an open neighbourhood of x. Wemay suppose that B is symmetric around x in the sense that B = 2x−B.Since x ∈ C, there exists y ∈ B ∩ C. Let z = 2x− y ∈ B ⊂ U0. For s ∈ I,we have 〈x, es〉 = 0, 〈y, es〉 > 0, and hence 〈z, es〉 < 0. Let w ∈ W suchthat z ∈ wC. By 1.2.2(b), I ⊂ {s ∈ S | ℓ(sw) < ℓ(w)}. By 1.3.7, I isspherical. �

3 The normalizer of a parabolic subgroup

3.1 The normalizer of a parabolic subgroup

We will describe a result of Deodhar, which will be used more than oncein this thesis (compare 6.8.1). In this subsection, we use it to reduce theconjugacy problem in Coxeter groups to the case Pc(v) = Pc(w) = W(3.1.13), assuming we know how to compute parabolic closures.

The part of this subsection before 3.1.6 follows Cohen [20]. Probably,the subsequent results are also known, but perhaps not published.

Recall our notation ΠI = {es | s ∈ I}. We note wIΠI = −ΠI forspherical I ⊂ S.

3.1.1 Definition. Let I ⊂ S, s ∈ S − I. Write K for the connected compo-nent of I ∪ {s} containing s. If K is spherical, then we define

ν(I, s) = wK−{s}wK .

Otherwise, ν(I, s) is not defined.

We haveν(I, s)−1ΠI = ΠJ

for some J = (I ∪ {s}) − {t}, t ∈ K.Consider the directed labelled graph K whose vertices are the subsets

of S, and in which there is an edge from I to J , labelled s, each time ν(I, s)exists, and ν(I, s)−1ΠI = ΠJ .

3.1.2 Proposition. If Is→ J is a labelled edge of K, then so is J

t→ I,where either {t} = I − J , or I = J and t = s.

Proof. Let Is→ J be a labelled edge. Then ν(I, s)−1es = −et for some

t ∈ K and ν(I, s)−1eu = ef(u) for some bijection f : I → J . It is easy to seethat ν(J, t) is defined. Write I ′ = (J∪{t})−{s}. Then ν(J, t)−1et = −er forsome r ∈ K := I∪{s}, and ν(J, t)−1eu = eg(u) for some bijection g: J → J ′.


It follows that ν(I, s) ν(J, t) ∈ GK ∩WK = {1}. Hence ν(J, t) = ν(I, s)−1.This proves that

Jt−→ I

is an edge of K. Since J = (I ∪ {s}) − {t}, we have either {t} = I − J orI = J and t = s. �

3.1.3 Theorem (Deodhar). Let I, J ⊂ S, w ∈W be such that w−1ΠI =ΠJ . Then there exists a directed path

I = I0s0−→ I1

s1−→ · · · st−→ It+1 = J

in K such thatw = ν(I0, s0) · · · ν(It, st)

and

ℓ(w) =t∑

i=0

ℓ(ν(Ii, si)).

Proof. [27, prop. 5.5]. For finite Coxeter groups, it was proved by Howlett[35, Lemma 4]. �

For the rest of this subsection, fix I ⊂ S. Let K0 be the connectedcomponent of K containing I. Let T be a spanning tree of K0. For J ∈ T ,let

µ(J) = ν(I0, s0) · · · ν(It, st)

whereI = I0

s0−→ I1s1−→ · · · st−→ It+1 = J

is the unique non-reversing path in T from I to J . For each labelled edgee = (I0

s0→ I1), let λ(e) = µ(I0)ν(I0, s0)µ(I1)−1. Note λ(e)ΠI = ΠI .

3.1.4 Definition. Define the group GI = {w ∈W | wΠI = ΠI}.

Note GI ⊂ DII by 1.2.2(b). We have GI ∩WI = {1}.

3.1.5 Corollary. GI is generated by all λ(e) where e runs through theedges of K0 which are not an edge of T .

Proof. This follows immediately from 3.1.3 and the fact that π1(K0, I) isgenerated by the paths in K0 from I to I containing exactly one edge notin T . �

3.1.6 Proposition. Let I, J ⊂ S, g ∈ D∅J such that g−1WIg = WJ .Then ΠI = gΠJ . In particular, g ∈ DIJ .

Proof. By 1.4.3, there exists a unique set X ⊂ Φ+ such that(1) {rα | α ∈ X} generates WI .(2) For all different α, β ∈ X:

(α, β) ∈ {− cos(π/m) | m ∈ Z>1} ∪ (−∞,−1].

Note that X is nothing else but ΠI . We will show that gΠJ shares theseproperties with ΠI . Since g ∈ D∅J , we have gΠJ ⊂ Φ+. Since g−1WIg =WJ , the set gΠJ satisfies (1). It obviously satisfies (2) as well. By unicityof X, we conclude gΠJ = ΠI . �

26 Daan Krammer

Figure 2: The Coxeter graph considered in 3.1.10

bc

bc bc

bc

bc

bc

1

2 3

4

5

6

3.1.7 Corollary. Let I, J ⊂ S. Then WI and WJ are conjugate in W ifand only if I and J are in the same connected component of K.

Proof. ‘If’ is easy. In order to prove ‘only if’, let g−1WIg = WJ . Byreplacing g by gw for an appropriate w ∈ WJ , we may suppose g ∈ D∅J .By 3.1.6, g−1ΠI = ΠJ . By 3.1.3, I and J are in the same connectedcomponent of K. �

We write N(WI), Z(WI) for the normalizer and centralizer, respec-tively, in W of WI .

3.1.8 Definition. Denote the natural group homomorphism GI → Sym(I)by φ. By ψ we denote the (unique) map N(WI) →W such that {ψ(w)} =wWI ∩D∅,I .

3.1.9 Proposition.

(a) The group N(WI) equals the semi-direct product GI ⋉WI , the nat-ural projection N(WI) → GI being ψ.

(b) ker φψ ⊂ Z(WI)WI .

Proof. (a). Note GI ∩ WI = 1, and that GI normalizes WI . Thus GI

and WI generate a semi-direct product GI ⋉ WI , which we denote by N .We will prove N = N(WI). ⊂ is clear. ⊃. Let w ∈ N(WI). Write{g} = wWI ∩D∅,I . Note g ∈ N(WI). Thus, by 3.1.6, g ∈ GI . This provesN = N(WI). From the foregoing, it also follows that ψ is the naturalprojection.

(b). Let w ∈ ker φψ. Then ψ(w)es = es for all s ∈ I, which impliesψ(w) ∈ Z(WI). Hence w ∈ ψ(w)WI ⊂ Z(WI)WI . �

3.1.10 Remark. The reverse inclusion of 3.1.9(b) does not necessarily hold.It is well known that for the Coxeter group of type An, (with generat-ing set S = {1, . . . , n}, and mst = 2 if |s − t| > 1 and mst = 3 if|s − t| = 1), the longest element maps es to −en+1−s. Using this fact,we will give a counterexample to the reverse inclusion. Let (W,S) be theCoxeter system given by the Coxeter graph of Figure 2, where we denoteS = {1, . . . , 6}. Define subsets I, J,K ⊂ S by I = {1, 2}, J = {1, 2, 3, 4, 6},K = {1, 2, 4, 5, 6}. Let w = wJwKwI . We have we1 = −e1, we2 = −e2,whence w ∈ Z(WI). Furthermore, wwI = wJwK interchanges e1 and e2.This shows that wwI = ψ(w) and w 6∈ ker φψ.

3.1.11 Corollary. Z(WI)WI has finite index in N(WI), and there exists


an algorithm finding g1, . . . , gk ∈ N(WI) such that

N(WI) = ⊔i giZ(WI)WI .

Proof. Finiteness of the index follows immediately from 3.1.9(b) and thefact that Imφψ ⊂ Sym(I) is finite.

The algorithm is as follows. By 3.1.5, we know how to find generatorsh1, . . . , hℓ of GI . Next, compute fi = ψhi. Write F = φGI . Then F isgenerated by the fi and is contained in the finite group Sym(I). Hence itis possible to compute the elements of F . By remembering how elementsof F are written as products of the fi, we find ki ∈ GI with the propertyGI = ⊔iki ker φ. Consequently, N(WI) = ⊔iki ker φψ. Since ker φψ ⊂Z(WI)WI , we have N(WI) =

⋃i kiZ(WI)WI . Lemma 3.1.12 enables us to

compute a maximal set X ⊂ {1, 2, . . . , ℓ} such that for all different i, j ∈ X,we have k−1

i kj 6∈ Z(WI)WI , that is, N(WI) = ⊔i∈XkiZ(WI)WI . �

3.1.12 Lemma. Let g ∈ N(WI). Then g ∈ Z(WI)WI if and only if foreach connected component J of I, we have either ψ(g)es = es for all s ∈ Jor J is spherical and ψ(g)es = −wJes for all s ∈ J .

Proof. Since ψ(g) ∈ gWI , we have g ∈ Z(WI)WI ⇔ ψ(g) ∈ Z(WI)WI .Hence we may suppose ψ(g) = g, that is, g ∈ GI . The ‘if’ part is easy. Inorder to prove the ‘only if’ part, suppose g ∈ GI ∩ Z(WI)WI , say g = zw,z ∈ Z(WI), w ∈WI . For all s ∈ I, we have zes = λses where λs ∈ {−1, 1}.For all s, t ∈ S we have (es, et) = (zes, zet) = λsλt(es, et). If s and t areadjacent in the Coxeter graph (that is, mst > 2), it follows that λs = λt.Hence λs = λt whenever s and t are in the same connected component ofS. Let J ⊂ I be a connected component of I, and let w be the projectionof w ∈ WI on WJ . Then gΦJ = zwΦJ = zΦJ = ΦJ . Since also g ∈ GI ,it follows that g ∈ GJ . The first case to be considered is zes = es for alls ∈ J . Then z ∈ GJ , so w = z−1g ∈ GJ , and w ∈ GJ ∩WJ = {1}. Itfollows that for all s ∈ J , we have ges = zwes = zes = es. The second caseis zes = −es for all s ∈ J . Then wΠJ = −ΠJ . By 1.2.6, WJ is finite andw = wJ . It follows that ges = zwes = zwJes = −wJes. �

3.1.13 Corollary. The conjugacy problem for Coxeter groups can be solvedas soon as for all Coxeter groups, the following two problems have beensolved.

(1) Given w ∈W , determine I ⊂ S, g ∈W such that Pc(w) = gWIg−1.

(2) Given v, w ∈W such that Pc(v) = Pc(w) = W , decide whether v, ware conjugate in W .

Proof. The algorithm is as follows. First use (1) to compute I, J, g, h suchthat Pc(v) = gWIg

−1, Pc(w) = hWJh−1. If I and J are not in the same

connected component of K, then v and w are certainly not conjugate, by3.1.7. If I and J are in the same connected component of K, we mayconjugate v and w so as to ensure Pc(v) = Pc(w) = WI . Next, compute{gi} such that N(WI) = ⊔igiZ(WI)WI (3.1.11). We claim that v and ware conjugate if and only if v and g−1

i wgi are conjugate in WI for some i.

‘If’ is trivial. Suppose v = g−1wg. Then WI = Pc(v) = Pc(g−1wg) =g−1Pc(w)g = g−1WIg, that is, g ∈ N(WI). Hence we may write g = gizh,z ∈ Z(WI), h ∈ WI . Hence v = h−1z−1g−1

i wgizh = h−1g−1i wgih, that is,

28 Daan Krammer

v and g−1i wgi are conjugate in WI . This problem was assumed solvable

in (2). �

3.2 Finite subgroups of Coxeter groups

In this subsection, we give a solution to the conjugacy problem for torsion(that is, finite order) elements of a Coxeter group. The results in thissubsection are well known. The reflection representation of a Coxeter groupsupplies a test whether a given element of the group is torsion or not.

3.2.1 Proposition.

(a) Let H ⊂ W be a finite subgroup. Then Pc(H) is finite. Moreover,H ′′ = Pc(H).

(b) There exists an algorithm which, given a finite subgroup H ⊂ W ,determines g, I such that Pc(H) = gWIg

−1.

Proof. (a). Choose any x ∈ U0. Let y =∑

h∈H hx. Then y is H-invariant,that is, H ⊂ y′. But since y ∈ U0, y′ is a finite parabolic subgroup by 2.1.4and 2.2.5, thus proving the first statement. The latter statement followsfrom the fact that spherical subsets of S are facial (2.2.4).

(b). The proof of (a) suggests the following algorithm. Consider theclassical root basis. Choose any x ∈ U0, for example, x =

∑s∈S fs. Com-

pute y =∑

h∈H hx, so that Pc(H) ⊂ y′. Since y ∈ U , it is possible tofind g ∈ W , I ⊂ S such that y ∈ gCI , which implies y′ = gWIg

−1. Sincey ∈ U0, y′ is finite. Since intersections of parabolic subgroups are againparabolic subgroups, Pc(H) is a parabolic subgroup of the Coxeter groupy′. Hence, the problem to find Pc(H) is now a finite one. �

3.2.2 Proposition. There exists an algorithm deciding whether two tor-sion elements in W are conjugate.

Proof. The proof is similar to that of 3.1.13. Let v, w ∈ W be torsion.By 3.2.1(b), we know how to compute I, J, g, h such that Pc(v) = gWIg

−1,Pc(w) = hWJh

−1. By 3.1.7, we can decide whether WI and WJ are con-jugate. If they are not, then v and w are certainly not conjugate. Nowassume Pc(v) = Pc(w) = WI . Let N(WI) = ⊔igiZ(WI)WI . Now v and ware conjugate in W if and only if there exists i such that v and g−1

i wgi areconjugate in WJ , which is a finite problem. �

Combination of 3.2.1 and the Galois correspondence gives the followingresult.

3.2.3 Lemma.

(a) Let F ⊂ Φ be a finite root subsystem, such that F = Span(F ) ∩ Φ.Then F is a parabolic root subsystem.

(b) Let P ⊂ E be a positive definite linear subspace such that P =Span(P ∩ Φ). Then P ∩ Φ is a finite parabolic root subsystem.

Proof. (a). Write H = 〈rα | α ∈ F 〉. We may assume H ′′ = W andE = Span Π. Note that H is finite by 1.2.6. Hence so is W , by 3.2.1. Sinceall mst are finite, our root basis is the classical root basis modulo somesubspace. Again by 1.2.6, our root basis is the classical one, and U = E∗.


Let Ann denote the annihilator maps for E and E∗. We have

SpanF = Ann AnnF = AnnH ′

= AnnH ′′′ = AnnW ′ = Ann {0} = E,

whence F = Φ ∩ SpanF = Φ.(b). By 1.2.5, the root system Φ ∩ P is finite. The result now follows

from (a). �

4 Hyperbolic reflection groups

The conjugacy problem for affine Coxeter groups is treated in subsec-tion 4.2. In the subsequent subsections, we will solve the conjugacy problemfor the so-called hyperbolic reflection groups. Subsection 4.3 is devoted tothe definitions of the involved notions.

4.1 Signatures

A real vector space equipped with a quadratic form is said to have signa-ture (k, ℓ,m) if it is isomorphic to Rk+ℓ+m with the quadratic form

Q(x1, . . . , xk, y1, . . . , yℓ, z1, . . . , zm) = (x21 + · · · + x2

k) − (y21 + · · · + y2

ℓ ).

By Sylvester’s theorem, k, ℓ and m exist uniquely. By signature (k, ℓ) wemean signature (k, ℓ, 0). It is easy to prove that a real vector space with aquadratic form of signature (k, ℓ,m) has a subspace of signature (k′, ℓ′,m′)if and only if k′ ≤ k, ℓ′ ≤ ℓ, k′ +m′ ≤ k +m and ℓ′ +m′ ≤ ℓ+m.

4.2 Affine Coxeter groups

An affine Coxeter group is an infinite Coxeter group associated to a rootbasis (E, (·, ·),Π) such that E has signature (n, 0, 1) for some n.

It is known that every irreducible affine Coxeter group is what is knownas an affine Weyl group, [12, Prop. 2, p. 146]. In particular, it is a semi-direct product Zn ⋊ W for some Weyl group W , which is a finite Coxetergroup itself. We remark here that the product of two affine Coxeter groupsis again affine. The easiest proof is by using the semi-direct decompositionsas above — the direct sum of the two root bases does not work.

4.2.1 Proposition. The conjugacy problem for affine Coxeter groups issolvable in linear time.

Proof. Let W be an affine Coxeter group. As noted above, we have a semi-direct decomposition W = T ⋊ W , T ∼= Zn, W a finite (Coxeter) group.

Moreover, an isomorphism T → Zn and a set of representatives R of W/Tcan be computed. Thus, it remains to check if there are t ∈ t, r ∈ R suchthat

tvt−1 = rwr−1. (6)

Since R is finite, we may suppose r to be fixed. Write v = t1r1, rwr−1 =

t2r2, ti ∈ T , ri ∈ R. We have tvt−1 = tt1(r1t−1r−1

1 )r1 = (tr1t−1r−1

1 )t1r1,so (6) is equivalent to r1 = r2 and

(tr1t−1r−1

1 )t1 = t2. (7)

30 Daan Krammer

Consider the homomorphism f : T → T , t 7→ tr1t−1r−1

1 . Equation (7) issolved by computing the image of f and checking whether it contains t2t

−11 .

As to complexity, computation of t1, r1, t2, r2 can be done in linear time;since there are only finitely many maps of the form f above, we may supposethem to be computed once and for all. More precisely, for a given f , thereare group homomorphisms pi: T → Z/di such that f = ∩i ker pi. Thisshows that a membership test for ker f is logarithmic in time. Thus, thecomplexity is linear. �

4.3 Hyperbolicity and word hyperbolicity

A hyperbolic reflection group is a Coxeter group associated to someroot basis (E, (·, ·),Π) such that E has signature (n, 1) for some n. Thisdefinition differs from the definitions used by many authors in that we donot require the group to have finite covolume in hyperbolic space. Themethods of this section apply to all hyperbolic reflection groups in oursense.

Roughly speaking, a group is called word hyperbolic if it admits a finitepresentation such that every word mapping to the identity in the group,contains strictly more than half of a relation. We will now state this defi-nition more precisely.

Let G be a group and let X ⊂ G be a generating subset. Let F (X)denote the free group on X. Let π: F (X) → G denote the unique homo-morphism with π(x) = x for all x ∈ X. Let ℓ: F (X) → N denote the lengthfunction with respect to X.

A group G is called word hyperbolic if it admits a finite presentation〈X,R〉 such that, with the above notation, for all w ∈ π−1(1), there area, b, p, q ∈ F (X) such that ab ∈ R, ℓ(ab) = ℓ(a)+ℓ(b), ℓ(b) < ℓ(a), w = paq,ℓ(w) = ℓ(p) + ℓ(a) + ℓ(q).

There exist equivalent definitions of word hyperbolicity, which relate thegeometry of the Cayley graph of G to the geometry of the hyperbolic plane.For these definitions of word hyperbolicity, we refer to [1], [22]. There itis also shown that word hyperbolicity does not depend on X, that is, if Gis word hyperbolic, then for every finite generating set X a presentation〈X,R〉 as above exists.

The following theorem classifies word hyperbolic Coxeter groups.

4.3.1 Theorem (Moussong). Let (W,S) be a Coxeter system, with Sfinite. The following are equivalent.

(1) W is word hyperbolic.(2) W has no subgroups isomorphic to Z2.(3) There is no I ⊂ S such that WI is irreducible affine of rank at least

3, and there are no disjoint I, J ⊂ S such that the subgroups WI

and WJ commute and are infinite.

Proof. [43, theorem 17.1]. Note that (2)⇒(3) is easy. We also mentionthe easy result that word hyperbolic groups have no subgroups isomorphicto Z2, whence (1)⇒(2). �

4.3.2 Remark. A hyperbolic reflection group is not necessarily word hyper-bolic, nor does the converse implication necessarily hold, namely, that every


Figure 3: The Coxeter graph considered in 4.3.2

bc

bc

bc bc

word hyperbolic Coxeter group is isomorphic to some hyperbolic reflectiongroup. A counterexample to the first implication is the group given bythe Coxeter graph of Figure 3. It is not word hyperbolic since it containsthe affine group of type A2. However, the quadratic form associated toits classical representation has signature (3, 1). Two counterexamples forthe converse implication (with the beautiful additional property that theDavis-Moussong complex is a topological manifold) are given by Moussong[43, p. 47].

In [31] it is proved that there is a linear solution to the conjugacyproblem for word hyperbolic groups.

4.4 Removed

This subsection has been removed from the original thesis, but its numberis retained here for easy reference.

4.5 Hyperbolic space

Let E be a real vector space equipped with a symmetric bilinear form (·, ·)of signature (n, 1). Sometimes we write Q(x) = (x, x). Define H = {x ∈E | (x, x) < 0}. Note that H has two connected components; we denotethese by H+, H−. Also note that H+ and H− are convex. Let S = {x ∈ E |(x, x) = 0, x 6= 0}, which has two components S+ = S∩H+, S− = S∩H−.Let O(n, 1) = {g ∈ GL(E) | (gx, gy) = (x, y) for all x, y ∈ E}. DefineO+(n, 1) to be the subgroup of O(n, 1) of all elements which preserve H+.Clearly, O(n, 1) = O+(n, 1)×{1,−1}. Define PSn = {R>0x | x ∈ E−{0}},and let π: E − {0} → PSn be the projection. Now hyperbolic n-spaceis defined to be Hn = πH+ ⊂ PSn. We call Sn−1 = πS+ the (n − 1)-sphere. It is the boundary in PSn of hyperbolic n-space. Elements ofSn−1 are called points at infinity.

The group O+(n, 1) acts transitively on Hn, and the stabilizer of anypoint is isomorphic toO(n). There exists anO+(n, 1)-invariant Riemannianmetric on Hn, which is unique up to multiples. For x, y ∈ H+, there existsa unique shortest path from πx to πy. It is π[x, y], which is simply denotedby [πx, πy]. The length of this path is called the distance between πx andπy, notation d(πx, πy). A formula for d(πx, πy) is:

d(πx, πy) = | log r|,

where r is either of the two (real, positive) roots of

4(x, y)2

(x, x)(y, y)= r + 2 + r−1.

32 Daan Krammer

Although hyperbolic geometry is happening in πH+, we prefer to considerH+. Having this in mind, we write d(x, y) = d(πx, πy) for x, y ∈ H+.

Since we are going to solve the conjugacy problem for reflection groupsin O+(n, 1), it is natural to classify conjugacy classes in O+(n, 1) first.

4.5.1 Proposition. Let g ∈ O+(n, 1). Then precisely one of the followingstatements is true.

(a) g has a fixed point in Hn.(b) g has no fixed points in Hn, and has exactly one fixed point at

infinity.(c) g has no fixed points in Hn, and has exactly two fixed points at

infinity.

Proof. [5, Prop. A.5.14]. �

4.5.2 Definition. In the situation of 4.5.1, g is called elliptic, parabolic,hyperbolic when (a), (b), (c), respectively, holds.

4.6 Hyperbolic reflection groups

For the rest of this section, let (E, (·, ·),Π) be a root basis of signature (n, 1),so that W is a hyperbolic reflection group. Probably, the assumption thatΠ is finite is not necessary for many results.

Since E is non-degenerate, it may be identified with E∗ by x 7→ (x, ·).We retain the notations of subsections 1.2 and 4.5.

4.6.1 Proposition. The set U∩H equals one of the components H+, H−.

Proof. In [41, cor. 1.3, p. 82] it is proved that U contains one of thecomponents. By 2.1.6, U does not meet the other component. �

In view of the above proposition, assume from now on U ∩H = H+.

4.6.2 Corollary. There exists an algorithm that, when given an elementof O+(n, 1), decides whether it is in W .

Proof. Let g ∈ O+(n, 1) be given. Fix z ∈ C ∩H+. Now gz ∈ H+ ⊂ U .Hence there exists w ∈W such that gz ∈ wC. By 1.2.4 we can find such aw. We claim that g ∈ W if and only if g = w. In order to prove ‘only if’,let g ∈W . Since gz ∈ wC ∩ gC, we have, by 1.2.2(c), g = w, which provesthe claim. This condition is easy to verify, which finishes the algorithm. �

4.6.3 Question. An open question is how to extend 4.6.2 to every root ba-sis (E, (·, ·),Π), that is, how to decide, when given an element g ∈ GL(E),whether g ∈ W . Let G ⊂ GL(E) be the subgroup of matrices of determi-nant ±1. Let us fix a basis of E, and for g ∈ G, let us denote by N(g) themaximum of the absolute values of the entries of the matrix of g with re-spect to this basis. Since W is a discrete subset of G, and since N−1([0, t])is compact for all t ∈ R, there exists a function f : R → R such that forall g ∈ W we have ℓ(g) ≤ f(N(g)). Hence, our question has affirmativeanswer if a computable function f with the above property exists.

4.6.4 Definition. Define DI = CI ∩H+, D = D∅ = C ∩H+.


4.6.5 Proposition. Let I ⊂ S. Then DI 6= ∅ ⇔ I is spherical.

Proof. ⇒. Let x ∈ DI . Then WI is a discrete subgroup of the stabilizer inO+(n, 1) of x. Since (x, x) < 0, this stabilizer is isomorphic to O(n), thatis, a compact group. Hence WI is finite. ⇐. Choose x ∈ D = C ∩ H+.Then qIx ∈ CI by 2.2.3. Since H+ is convex, and by 2.2.1, qIx ∈ H+.Hence qIx ∈ DI . �

4.7 The conjugacy problem

Retain the setting of subsection 4.6.

4.7.1 Definition. We define the map φ: U → C by {φ(x)} = Wx ∩ C.

4.7.2 Definition. Let g ∈ O+(n, 1) be parabolic (see 4.5.2). By Rg we willdenote the intersection of S+ with the eigenspace corresponding to the fixedpoint at infinity. Strictly speaking, Rg is the fixed point at infinity. Forhyperbolic g, the axis is defined to be the set of x ∈ Hn such that d(x, gx)is minimal.

4.7.3 Proposition. Let w ∈W be a parabolic element. Then Rw ⊂ U .

Proof. The proof consists of three steps.

Step 1: For all ε > 0 there exists x ∈ H+ such that d(x,wx) < ε. Proof:with respect to a certain basis of E, we have

w =

1 2 1 00 1 1 00 0 1 00 0 0 A

,

Q(x1, . . . , xn) = (x22 − x1x3) + (x2

4 + · · · + x25).

Let x = (a, 0, 1, 0, . . .). Then wx = (a + 1, 1, 1, 0, . . .). We have (x, x) =−a, (x,wx) = −(2a + 1)/2, (wx,wx) = (x, x) = −a. By subsection 4.5,d(x,wx) = | log r|, where r is either of the two roots of

4(x,wx)2

(x, x)(wx,wx)=

(2a+ 1

a

)2= r + 2 + r−1.

Clearly, if a approaches infinity, then r approaches 1, which proves step 1.

Step 2: For I ⊂ S, define

ε(I) = inf{∑

s∈I

d(x,Ds)∣∣∣ x ∈ D

}.

Here, Ds = D{s} from 4.6.4. If ε(I) = 0 then there exists y ∈ CI with(y, y) ≤ 0, y 6= 0. Proof: Let ω be the invariant Riemannian metric on Hn.Let η be any Riemannian metric on a neighbourhood in PSn−1 of Hn suchthat η ≤ ω on Hn. Such an η exists, because some model of Hn looks asfollows [4, p. 49]:

Hn = {x ∈ Rn : |x| < 1} ⊂ Rn ∪ {∞} = PSn, ω =|dx|

1 − |x|2 .

34 Daan Krammer

Thus, η may be chosen to be η = |dx| ≤ ω. Let d0 be the correspondingdistance. We have d0(x, y) ≤ d(x, y) for all x, y ∈ Hn.

Now suppose ε(I) = 0. For each integer k > 0, choose xk ∈ π(D) suchthat ∑

s∈I

d(xk, π(Ds)) ≤1

k.

Let K be the closure in PSn−1 of π(D). Since K is compact, the xk havea limit point x ∈ K. Since

∑

s∈I

d0(xk, π(Ds)) ≤∑

s∈I

d(xk, π(Ds)) ≤1

k,

it follows that ∑

s∈I

d0(x, π(Ds)) = 0,

in other words, x ∈ π(Ds) for all s ∈ I. Hence y has the desired propertieswhenever x = R>0y, which concludes step 2.

Step 3: End of the proof. Let

ε =1

#Smin

{ε(I)

∣∣∣ I ⊂ S, ε(I) 6= 0}.

Let x ∈ H+ be such that d(x,wx) < ε (existing by step 1). After conjugat-ing w, we may suppose x ∈ C. Consider the curve φ[x,wx], where φ is themap defined in 4.7.1. This is a closed curve in C of length d(x,wx). Let

I = {s ∈ S | φ[x,wx] ∩Ds 6= ∅}.

Note that w ∈WI . Furthermore, we have ε(I) = 0 since otherwise

ε(I) ≤∑

s∈I

d(x,Ds) ≤ (#I)d(x,wx) < (#I)ε ≤ (#S)ε ≤ ε(I)

which is a contradiction. By step 2, there exists y ∈ CI with (y, y) ≤ 0,y 6= 0. Since y ∈ CI and w ∈ WI , we have wy = y. But in 4.5.1,parabolic elements w of O+(n, 1) were characterized as elements having nofixed points in Hn, and exactly one in Sn−1, which is Rw. Hence Rw =R>0y ⊂ CI ⊂ U . �

4.7.4 Remark. A result analogous to proposition 4.7.3 is known for everydiscrete group in O+(2, 1) (that is, not only reflection groups), see [4, Cor.9.2.9, p. 216]. Probably, this result generalizes to every discrete group inO+(n, 1).

4.7.5 Definition. Let K be a subfield of R. We say that a root basis(E, (·, ·),Π) is defined over K if there exists a basis x1, x2, . . . , xn of Esuch that (xi, xj) ∈ K for all i, j, and Π ⊂

∑iKxi.

4.7.6 Theorem. Let W be a finite rank hyperbolic reflection group, de-fined over a real number field. Then the conjugacy problem for W is solv-able.


Proof. For algorithms on calculations in number fields, see subsection 5.9.We will describe the algorithm that decides if v, w ∈ W are conjugate.First calculate the matrices of v and w and decide if they are conjugatein O+(n, 1). If not, we are done, so suppose they are. Next, decide whichclass v and w fall into, elliptic, parabolic or hyperbolic (see 4.5.2). Thesethree cases are treated separately.

Case 1: v and w are elliptic. Now v and w have finite order. To seethis, note that the topological closure in O+(n, 1) of wZ is compact. SincewZ is discrete, it follows that w has finite order, as has v. But for finiteorder elements, the conjugacy problem has been solved in 3.2.2.

Case 2: v and w are parabolic. By 4.7.3, Rv, Rw ⊂ U . Hence we canfind x, y ∈W , I, J ⊂ S such that Rv ⊂ xCI , Rw ⊂ yCJ . If I 6= J , v and ware not conjugate, so suppose I = J . After conjugating v and w by suitableelements of W , we may suppose Rv, Rw ⊂ CI . Note that v, w ∈ WI , andgvg−1 = w ⇒ gRv = Rw ⇔ g ∈ WI . Hence v and w are conjugate if andonly if they are conjugate in WI . Note that Span ΠI has signature (k, 0, 1)for some k, because ΠI ⊥ Rw and w ∈ WI is not elliptic. Hence WI is anaffine Coxeter group. Now case 2 is finished by 4.2.1.

Case 3: v and w are hyperbolic. The key here is the axis of an hyperbolicelement. This idea is very similar to our solution to the conjugacy problemfor the case Pc(v) = Pc(w) = W irreducible, infinite, non-affine, whichis treated in subsection 6.7. We refer to 6.7.5 for an adaptation to ourcase. �

5 The axis

5.1 Introduction

When solving the conjugacy problem for a group G, it is natural to lookfor the shortest elements in a conjugacy class C(g) (with respect to a finitegenerating subset X). A particular nice case is if C(g) contains straightelements h, that is, ℓ(hn) = |n| ℓ(h) for all n ∈ Z. Torsion (that is finiteorder) elements different from 1 are examples of elements not conjugate toany straight element.

In subsection 5.4 we define a metric dΦ on U0 such that for all x, y ∈W ,we have dΦ(xC, yC) = dW (x, y). The axis Q(w) is then defined to be theset of those x ∈ U0 such that d(x,wnx) = nd(x,wx) for all n > 0. Thisgeneralizes the straight elements. The advantage is that Q(w) is nonemptyfor all w ∈W .

The action of W on Φ is also important. For given w ∈ W , we willin subsection 5.2 classify roots into three classes, called w-periodic, w-evenand w-odd roots. In subsection 5.7, a refinement of this classification ismade.

Two applications are that the parabolic closure Pc(w) of w is generatedby the so-called w-essential reflections (5.8.3, see 5.7.5 for a definition ofessential), and a cubic algorithm computing the parabolic closure (5.10.9).

5.2 Periodic, even and odd roots

5.2.1 Definition. Let w ∈ W , and let A ⊂ W be a half-space. We call Aw-periodic or simply periodic if there exists n > 0 such that wnA = A.

36 Daan Krammer

5.2.2 Proposition. Let w ∈ W , and let A ⊂ W be a half-space. Thenexactly one of the following holds.

(1) A is periodic.(2) A is not periodic and for every x ∈ W , the number #{n ∈ Z | A

separates wnx from wn+1x} is finite and even.(3) A is not periodic and for every x ∈ W , the number #{n ∈ Z | A

separates wnx from wn+1x} is finite and odd.

Proof. Let A be a half-space which is not periodic.For x, y ∈W , let L(x, y) denote the set of n ∈ Z such that A separates

wnx from wny. We must show that L(x,wx) is finite and that the parityof #L(x,wx) does not depend on x (only A).

Note that L(x, y) is also the set of n ∈ Z such that w−nA separates xfrom y. But there are only finitely many half-spaces separating x from y.If L(x, y) were infinite, it would follow that there are distinct integers m,nsuch that w−mA = w−nA, contradicting the fact that A is not periodic. SoL(x, y) is finite for all x, y ∈W .

If X,Y are sets, write X⊕Y := (X−Y )∪ (Y −X). We have L(x, y) =L(y, x) and

L(x, y) ⊕ L(y, z) = L(x, z) for all x, y, z ∈W .

We find

L(x,wx) ⊕ L(y, wy)

=(L(x,wx) ⊕ L(wx, y)

)⊕

(L(wx, y) ⊕ L(y, wy)

)

= L(x, y) ⊕ L(wx,wy)

which has even cardinality because L(wx,wy) has the same cardinality asL(x, y). It follows that the parity of #L(x,wx) does not depend on x. �

5.2.3 Definition. In cases (2) and (3) in 5.2.2, we call A even and odd,respectively.

In order to express the relation to w, we will sometimes speak aboutw-even half-spaces, etc. The characterization of 5.2.2 is also valid for rootsand reflections.

5.2.4 Corollary. The characterization of 5.2.2 is conjugacy covariant, thatis, for all w, g,A, we have

A is w-even ⇔ gA is gwg−1-even,

and similarly for odd.

Proof. Write S(A,w, x) = {n ∈ Z | A separates wnx from wn+1x}. Itis readily seen that S(A,w, x) = S(gA, gwg−1, gx). Let A be even. Thenfor all x ∈ W , we have #S(gA, gwg−1, x) = #S(A,w, g−1x) is finite andeven. Hence gA is gwg−1-even. The converse and the case of odd A aresimilar. �

5.2.5 Examples. (a). For w of finite order, all half-spaces are periodic.


Figure 4: The affine Coxeter group bc bc bc

1 2 3

4 4

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1 1 1

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∅

w

(b). Consider the universal Coxeter group (W,S), that is, all mst

are infinite. Then the Cayley graph is a tree. Let w ∈ W be cyclicallyreduced, that is, w = s1s2 · · · sn, si ∈ S, si 6= si+1, sn 6= s1. Let xk ∈ W(k ∈ Z) be defined by xk = s1s2 · · · sk (1 ≤ k ≤ n) and xk+n = wxk for allk. Then a half-space is odd if and only if it is one of the two half-spacesseparating xk from xk+1 for some k. There are no periodic half-spaces.

(c). Consider the affine Coxeter group W of Figure 4. Elements ofW correspond to triangles, and walls with lines in the figure. The identityelement is marked ∅, and some edges {v, vs} have been labelled s. Considerthe element w = 12123. It is easy to see that w is a glide-reflection, thatis, a reflection through the thick line, followed by a translation over thevector v. All horizontal lines define periodic walls; the other walls are odd.We will come back to this example a few times.

5.2.6 Lemma. There are only finitely many wZ-orbits of odd half-spaces.

Proof. Let A be an odd half-space. Since odd integers are non-zero, thereexists n ∈ Z such that A separates wn from wn+1. Hence w−nA separates1 from w. But there are only finitely many half-spaces separating 1 fromw, which proves the lemma. �

5.2.7 Lemma. For all w ∈ W , n > 0, a half-space is w-odd if and only ifit is wn-odd, and similarly for even.


5.3 Distinguishing even roots from odd ones

Whereas in the previous subsection we considered half-spaces, from now onwe will prefer roots over half-spaces. The translation between these two isprovided by subsection 1.4, especially 1.4.7.

Write c = max(2,#{(α, β) | α, β ∈ Φ} ∩ (−1, 1)), r = #S. Note that c

38 Daan Krammer

is finite by 3.2.1(a).

A sequence a1, a2, . . . of complex numbers is said to satisfy a linearrecurrence of order k if there are p0, . . . , pk ∈ C (p0, pk 6= 0) such thatp0an + · · · + pkan+k = 0 for all n. If this is the case, then the sequence iscompletely determined by k consecutive entries and p0, . . . , pk.

5.3.1 Lemma. There exists a constant N = N(W ) such that the followingholds. For each w ∈ W , α ∈ Φ, there exists n such that 1 ≤ n ≤ N and|(α,wnα)| ≥ 1. We may take N = cr.

Proof. Write an = (α,wnα), and note a−n = an. Let K be the smallestpositive integer such that |aK | ≥ 1, or infinity if such an aK does notexist. Consider the r-tuples (an+1, an+2, · · · , an+r) such that −K < n+ 1,n + r < K. If all r-tuples are different, then, since there are at most cr

possible r-tuples, we have 2K − r ≤ cr, whence K ≤ (cr + r)/2. If twor-tuples are equal, then, since the sequence an satisfies a linear recurrenceof order r, the sequence is periodic. The period is at most cr, which showsthat for some n with 1 ≤ n ≤ cr, we have an = a0 = (α, α) = 1. In bothcases, we have |an| ≥ 1 for some n with 1 ≤ n ≤ max((cr+r)/2, cr) = cr. �

5.3.2 Proposition. There exists a constant N = N(W ) such that for allw ∈W , the following hold. Let α be w-odd. Then

(a) For all n ≥ 1: (α,wnα) > −1,(b) There exists n, 1 ≤ n ≤ N : (α,wnα) ≥ 1.

Let α be w-even. Then

(c) For all n ≥ 1: (α,wnα) < 1,(d) There exists n, 1 ≤ n ≤ N : (α,wnα) ≤ −1.

We may take N to be the constant of 5.3.1.

Proof. Write A = A(α). First, we prove (a) and (c). In order to prove (a),let α be odd, say wk ∈ A for k > 0 big enough. Then there exists k > 0 withwk, wk−n ∈ A. Hence A∩wnA 6= ∅, and, similarly, W−A and W−wnA arenot disjoint. In other words, neither of the sets A and W−wnA is containedin the other. By 1.4.7(b), we find (α,−wnα) < 1, which proves (a). In orderto prove (c), suppose α to be even but (α,wnα) ≥ 1. By 1.4.7(b), we have,after replacing α by −α if necessary, A ⊂ wnA. Since A is not periodic,there exists x ∈ wnA − A. Since · · · ⊂ w−nA ⊂ A ⊂ wnA ⊂ · · · , we findx ∈ wknA⇔ k > 0. Hence α is odd, a contradiction, which proves (c).

Now we will prove (b) and (d). Let α be either odd or even. Let n besuch that 1 ≤ n ≤ N and |(α,wnα)| ≥ 1. Such an n exists by 5.3.1. If αis odd, then by (a) we have (α,wnα) ≥ 1, which proves (b). If α is even,then by (c) we have (α,wnα) ≤ −1, which proves (d). �

We refer to 5.10.1 for an algorithm, based on 5.3.2, to decide whethera root is even or odd.

5.4 A pseudometric on U0

We turn attention to the interior U0 of U . By 2.2.5, it equals the union ofthe facets with finite stabilizer.


For a root α, let fα: U0 → {−1/2, 0, 1/2} be the map

fα(x) =

−1/2 if 〈x, α〉 < 0,0 if 〈x, α〉 = 0,1/2 if 〈x, α〉 > 0.

Recall that a pseudometric on a set X is a map d: X ×X → R≥0 suchthat for all x, y, z ∈ X one has d(x, z) ≤ d(x, y) + d(y, z). We definepseudometrics dα and dΦ on U0 by

dα(x, y) = |fα(x) − fα(y)|, dΦ(x, y) =∑

α∈Φ+

dα(x, y).

Let x, y ∈ U0. A root α is said to separate x from y if dα(x, y) > 0.We say that α strictly separates x from y if dα(x, y) = 1.

For x ∈ U0, the set {y ∈ U0 | dΦ(x, y) = 0} equals the facet containingx. Thus, the facets in U0 form a metric space. By 2.2.4 and 2.2.5, thismetric space is purely combinatorial, that is, up to isometry, it dependsonly on the Coxeter system, not on the root basis. In fact, the facets in U0

are in bijection with the union over the spherical I ⊂ S of W/WI , by theidentification wCI 7→ wWI .

5.4.1 Lemma.(a) For all v, w ∈W we have dΦ(vC,wC) = dW (v, w).(b) For all x, y ∈ U0, the distance dΦ(x, y) is finite.

Proof. (a). This follows from the fact that w 7→ wC is a bijection, and αseparates v from w if and only if it separates every point of vC from everypoint of wC.

(b). Let I ⊂ S be spherical. Then dΦ(CI , C) is finite (more precisely,it is half the number of reflections in WI). Using (a), dΦ(vCI , wCJ) ≤dΦ(CI , C) + dΦ(vC,wC) + dΦ(CJ , C) is finite. �

Recall µ(α) = {x ∈ U0 | 〈x, α〉 = 0} = K(α) −H(α). We write µ(A),H(A), K(A) for µ(α), H(α), K(α), where A = A(α).

The following lemma translates the language of the Tits cone into thelanguage of the Coxeter group, and will frequently be used. The proof isleft to the reader.

5.4.2 Lemma. Let g ∈W , and let A ⊂W be a half-space. The followingare equivalent.

(1) g ∈ A.(2) gC ⊂ H(A).(3) gC ⊂ K(A).(4) gC ⊂ K(A). �

5.5 The axis

We now come to the definition of an object that will be the center of studyin this section.

5.5.1 Definition. For each w ∈W , define the axis Q(w) to be

{x ∈ U0 | for all n ∈ Z: dΦ(x,wnx) = |n| dΦ(x,wx)}.

40 Daan Krammer

Figure 5: w = 123 in the Coxeter group bc bc bc

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42 Daan Krammer

Figure 7: w = 1234 in the Coxeter group bc bc bc bc

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5.5.2 Examples. (a). Q(w) contains the fixed points in U0 of w. It is easyto see that when there is at least one such fixed point (equivalently, w istorsion, by 2.2.5 and 3.2.1), then Q(w) equals the set of fixed points.

(b). Let us consider again the universal Coxeter group of 5.2.5(b).Then all facets have codimension 0 or 1. The facets of codimension 0 arein bijection with W , those of codimension 1 with the edges in the Cayleygraph. In the notation of 5.2.5(b), Q(w) equals (

⋃k∈Z xkC)∪(

⋃k∈Z xkCsk

).

(c). In the example of 5.2.5(c), Q(w) equals the thick line. The proofis left to the reader. The reader may wish to use (1)⇒ (2) of 5.5.4 below,applied to a root associated to the thick line.

(d). Let X ⊂ E∗ be a w-invariant two-dimensional subspace, on whichw has positive real eigenvalues. ThenX∩U0 is contained inQ(w). A partic-ular case is where W is affine, w ∈W a translation, and X = Span{x,wx}for some x ∈ U0. It follows that in this case, Q(w) = U0.

(e). In figures 5, 6 and 7, we give a picture of Q(w) for three cases,following the conventions of 5.2.5(c). All three examples have the prop-erty that W can be made to act (cocompactly) discretely on the hyperbolicplane. Otherwise, drawing pictures would become difficult. The fundamen-tal regions are triangles (in figures 5 and 6) and a 4-gon (in figure 7). The4-gon has three right angles and an angle equal to π/3. The pictures areconstructed in such a way that the action of w on the plane of the pictureis a translation or a glide-reflection, according to whether ℓ(w) is even orodd. Moreover, we have drawn the pictures in such a way that Q(w) is anopen strip, as shown in the pictures. As an example, one wall is dottedin each figure. It has the property that one of the associated roots α isminimal under the condition Q(w) ⊂ H(α).

Denote the centralizer in W of w by Z(w).

5.5.3 Lemma. For all g, w ∈ W , we have Q(gwg−1) = g Q(w). In partic-ular, Q(w) is invariant under Z(w).


5.5.4 Lemma. Let w ∈W , x ∈ U0. The following are equivalent.

(1) x ∈ Q(w).(2) For every α ∈ Φ, the function Z → {−1/2, 0, 1/2}, n 7→ fα(wnx) is

monotonic.

Proof. Statement (1) is equivalent to saying that for all k, ℓ,m ∈ Z, k <ℓ < m, we have

dΦ(wkx,wmx) = dΦ(wkx,wℓx) + dΦ(wℓx,wmx).

This is equivalent to saying that for all α ∈ Φ, k, ℓ,m ∈ Z, k < ℓ < m, wehave

dα(wkx,wmx) = dα(wkx,wℓx) + dα(wℓx,wmx).

This, in turn, is equivalent to (2). �

The following lemma gives some more properties of the function of5.5.4(2).

44 Daan Krammer

5.5.5 Lemma. Let w ∈W , x ∈ U0, α ∈ Φ and write g: Z → {−1/2, 0, 1/2},g(n) = fα(wnx).

(a) Suppose that g is monotonic. If g is not constant, then its imagecontains {−1/2, 1/2}. Furthermore, g is not constant if and only ifα is odd.

(b) Suppose that α is not odd. Then g is monotonic if and only if it isconstant.

Proof. (a). Suppose g is not constant, but the image of g does not contain{−1/2, 1/2}. Then after replacing w by w−1 if necessary, there existsN ∈ Z

such that for all n > N , we have g(n) = 0, or, equivalently, 〈wnx, α〉 = 0.Since the sequence 〈wnx, α〉 (n ∈ Z) satisfies a linear recurrence, it followsthat 〈wnx, α〉 = 0 for all n, and g is constant, a contradiction. This provesthe first statement.

In order to prove the latter statement, write A = A(α) and choosey ∈W such that x ∈ yC. We have

g(n) = 1/2 ⇒ wny ∈ A, g(n) = −1/2 ⇒ wny 6∈ A. (8)

Suppose that g is not constant. Then (8) and the fact that g is monotonicimmediately imply that A is odd. Now suppose g to be constant but A odd.Then from (8) it follows that g is constant zero. Hence the walls µ(wnα)all pass through x ∈ U0. But there are only finitely many walls througha point in U0. Hence α is periodic, contradicting the fact that A is odd.This concludes the proof of (a).

(b). Easy and left to the reader. �

5.5.6 Definition. Let α be a root. Then we call α outward if for some(hence all) x ∈ U0 the following holds. For almost all n ∈ Z, we haven〈wnx, α〉 < 0. It is called inward if −α is outward. Similarly for half-spaces. By 5.2.2, α is odd if and only if α or −α is outward. The set ofoutward roots is denoted Out(w). We define r(w) to be #(wZ\Out(w)).So r(w) is finite by 5.2.6.

The following proposition gives another characterization of Q(w). Ourmembership test for Q(w) (5.10.5) will be based on it.

5.5.7 Proposition. For all x ∈ U0, we have dΦ(x,wx) ≥ r(w), withequality if and only if x ∈ Q(w). In particular, r(w) ≤ ℓ(w).

Proof. Let x ∈ U0. Write r = r(w), and let α1, . . . , αr be outward rootssuch that Out(w) equals the disjoint union over i of wZαi. Then for everyoutward root α, by 5.5.5, we have

∑n∈Z dα(wnx,wn+1x) ≥ 1, whence

dΦ(x,wx) =∑

α∈Φ+

dα(x,wx) ≥r∑

i=1

∑

n∈Z

dαi(wnx,wn+1x) ≥ r(w).

Equality holds if and only if x ∈ Q(w), by 5.5.4 and 5.5.5. �

5.5.8 Lemma. Let α be even. Then Q(w) ⊂ H(α) or Q(w) ⊂ H(−α) (orboth).


Proof. Write A = A(α). Let g ∈ W . By replacing α by −α if necessary,we may suppose wng ∈ A for almost all n ∈ Z. We will show Q(w) ⊂ H(α).Let x ∈ Q(w). Let h ∈ W be such that x ∈ hC. We will show wnh ∈ Afor almost all n ∈ Z. If not, infinitely many w−nA separate g from h.Thus, A is periodic, a contradiction. Hence, wnh ∈ A for almost all n ∈ Z.Hence, wnx ∈ K(α) for almost all n, by 5.4.2. By 5.5.4 and 5.5.5, it followsthat either wZx ⊂ H(α) or wZx ⊂ µ(α). In the latter case, µ(wnα) passesthrough x for all n ∈ Z. This implies that α is periodic, a contradiction.This proves Q(w) ⊂ H(α). �

5.5.9 Definition. Call a root α w-supporting if Q(w) ⊂ H(α). Similarlyfor half-spaces.

The above lemma says that for every even root α, one of α,−α issupporting. From the results of the following subsection, it will follow thatthe other one is not supporting.

5.6 Non-emptiness of the axis

In this subsection, we will prove that Q(w) is non-empty. It will be usefulto consider subsets Q1(w) ⊂ Q2(w) ⊂ Q(w) as follows.

5.6.1 Definition. Define Q2(w) to be the set of x ∈ Q(w) such that forall outward α, 〈x, α〉 ≤ 〈x,wα〉, and for all periodic α, 〈x, α〉 = 〈x,wα〉.Let Q1(w) = Q2(w) ∩ E>0, where E>0 denotes the sum of the generalizedeigenspaces in E∗ of w with eigenvalue in R>0.

The set Q1(w) will be used in subsection 6.5. It is left to the reader tocheck the analogues to 5.5.3.

Whereas Q(w) is a purely combinatorial object (that is, it is a union offacets), Q1(w) and Q2(w) are not. On the other hand, we are able to provethat Q1(w) and Q2(w) are convex, which is only conjectural for Q(w) (see5.7.10).

5.6.2 Proposition. Q1(w) and Q2(w) are convex.

Proof. It is enough to prove that Q2(w) is convex, for it immediatelyfollows that Q1(w) is convex too. Let x, y ∈ Q2(w), and write z = x + y.It is enough to show that z ∈ Q(w), for it would then easily follow thatz ∈ Q2(w). For every root α, we must prove that n 7→ fα(wnz) is monotonic(5.5.4). For α even this follows from 5.5.8. For α periodic, n 7→ 〈z, wnα〉is constant zero by definition of Q2(w) whence so is n 7→ fα(wnz). Forα odd, n 7→ 〈z, wnα〉 is monotonic by definition of Q2(w), whence so isn 7→ fα(wnz). �

5.6.3 Lemma. There exists a constant N = N(W ) such that for all w ∈W , and every w-periodic root α, we have wNα = α.

Proof. We may suppose that E is the classical root base. Let K be thenumber field generated by {cos(π/mst) | s, t ∈ S}. Let EK ⊂ E be theK-vector space with basis {es | s ∈ S}. Note Φ ⊂ EK . Let w ∈W , and letα ∈ Φ be w-periodic. Let V ⊂ EK be the Q-vector space spanned by wZα.

46 Daan Krammer

There exists a w-invariant decomposition of Q-vector spaces V =⊕Vi and

natural numbers ni such that the characteristic polynomial of w|Viis the

ni-th cyclotomic polynomial. In particular, φ(ni) = dim Vi ≤ dim V ≤dimQ EK . Since limn→∞ φ(n) = ∞, the ni are bounded by a constant mdepending only on W . Putting N = lcm (1, 2, . . . ,m), we have wNα = α,which proves the theorem. �

5.6.4 Remark. A better bound for the constant N of 5.6.3 will be given in6.3.12. The proof of this result will depend indirectly on 5.6.3.

5.6.5 Theorem. There exists a constant N = N(W ) such that for allw ∈W , we have (1 + w + w2 + · · · + wN−1)U0 ⊂ Q2(w).

Proof. Let N0 be the constant defined in 5.3.2, so that for every root α,there exists k with 1 ≤ k ≤ N0 and |(α,wkα)| ≥ 1. Let N1 be the constantof 5.6.3. Let N2 = 2 lcm(1, 2, . . . , N0), N = lcm(N1, N2). We will show thatN has the desired property. Let x ∈ U0. Let y = (1 + w + · · · + wN−1)x.Note wy − y = wNx − x. By 5.5.4 and 5.5.5, it is enough to show that〈y, α〉 = 〈wy, α〉 for every periodic α, and 〈wy, α〉 ≤ 〈y, α〉 for every outwardα, and 〈wZy, α〉 ⊂ R>0 or R<0 for every even α.

Let α be a periodic root. By our choice of N1, wNα = α. Hence

〈wy − y, α〉 = 〈wNx− x, α〉 = 0.Next, let α be outward. By 5.3.2 and 1.4.7, there exists k, 1 ≤ k ≤ N0

with α < wkα. By 1.4.7(d), it follows that 〈wn+kx, α〉 ≤ 〈wnx, α〉 forall n ∈ Z. Since k|N , we find 〈wNx, α〉 ≤ 〈x, α〉. Hence 〈wy − y, α〉 =〈wNx− x, α〉 ≤ 0.

Finally, consider the case of even α. By 5.3.2, there exists k with 1 ≤k ≤ N0 and (α,wkα) ≤ −1. By 1.4.7, the set 〈U0, α+wkα〉 is contained inR>0 or R<0, say in R>0. We will show 〈y, α〉 > 0, whence 〈wZy, α〉 ⊂ R>0,since the same argument applies to wpy too. Since 2k divides N , we canpartition the set {0, 1, . . . , N − 1} into pairs, each having difference k. Foreach pair, say {n, n+ k}, we have 〈wnx+ wn+kx, α〉 > 0. Adding over allpairs, we find 〈y, α〉 = 〈(1 + w + · · · + wN−1)x, α〉 > 0. �

5.6.6 Corollary. For every w ∈ W , the limit limn→∞

ℓ(wn)/n exists and

equals r(w).

Proof. By 5.6.5, Q(w) is non-empty. Let x ∈ Q(w). Then for all n ∈ N,we have, by 5.4.1 and definition of Q(w),

|ℓ(wn) − ndΦ(x,wx)| = |dΦ(C,wnC) − dΦ(x,wnx)| ≤ 2 dΦ(C, x),

which shows that limn→∞

ℓ(wn)/n exists and equals dΦ(x,wx) = r(w). �

5.6.7 Corollary. We have dΦ(C,Q(w)) = O(ℓ(w)) for all w ∈ W . Moreprecisely, dΦ(C,Q(w)) ≤

(N2

)ℓ(w), where N is the constant of 5.6.5.

Proof. Let x ∈ C and put y = (1 + w + · · · + wN−1)x. Then y ∈ Q(w) by5.6.5. Furthermore, each root separating C from y separates C from oneof x,wx, . . . , wN−1x, and similarly for strict separation. Hence

dΦ(C, y) ≤N−1∑

k=0

dΦ(x,wkx) =N−1∑

k=0

ℓ(wk) ≤N−1∑

k=0

k ℓ(w) =(N2

)ℓ(w). �


Let P [X] denote the set of non-zero real polynomials in X with non-negative coefficients, P [X] = {a0 +a1X+ · · ·+anX

n ∈ R[X]−{0} | ai ≥ 0for all i}.

5.6.8 Lemma. Let z be a complex number. Then there exists f ∈ P [X]with f(z) = 0 if and only if z is not a positive real number.

Proof. ‘Only if’ is easy. To prove ‘if’, note that there exists n ∈ Z>0

such that Re(zn) ≤ 0. Now f = (Xn − zn)(Xn − zn) ∈ P [X] satisfiesf(z) = 0. �

5.6.9 Lemma. Let A ∈ GLn(R). Then there exists f ∈ P [X] such thatall eigenvalues of f(A) are in R≥0.

Proof. Let f0f1 ∈ R[X] be the characteristic polynomial of A, and assumef0(λ) = 0 ⇒ λ ∈ R>0 and f1(λ) = 0 ⇒ λ 6∈ R>0 and that f0 is monic. By5.6.8, there exists f ∈ P [X] such that f1 divides f . It is easy to see that fhas the desired property. �

5.6.10 Theorem. For every w ∈W , the set Q1(w) is non-empty.

Proof. By 5.6.9, there exists f ∈ P [X] such that all eigenvalues of f(w) arein R≥0. By 5.6.5, Q2(w) is non-empty; let y ∈ Q2(w). Let z = f(w)y. Thenz ∈ Q2(w) since Q2(w) is convex. Moreover, z ∈ E>0 by our constructionof f . Hence z ∈ E>0 ∩Q2(w) = Q1(w). �

5.7 Critical roots

In this subsection, we fix an element w ∈W .

5.7.1 Lemma. Let x ∈ U0, N > 0. Then there are only finitely manyroots α with 0 ≤ 〈x, α〉 ≤ N .

Proof. We may suppose E = Span(Π). First we prove the assertion for thespecial case of x being on no wall. After translation by an element of W ,we may suppose x ∈ C. Since C is open, there exists a basis {x1, . . . , xr}of E such that xi ∈ C for all i, and x = x1 + · · · + xr. Write

X = {α ∈ Φ | 0 ≤ 〈x, α〉 ≤ N}.

Note X ⊂ Φ+. Thus, X is a subset of the following compact set: {y ∈E | 〈xi, y〉 ≥ 0 for all i,

∑i〈xi, y〉 ≤ N}. Moreover, X is a discrete set by

1.2.5. We conclude that X is finite, which proves the case x on no wall.

Now we will solve the general case. There exist y, z ∈ U0 on no wall,such that x = y + z. Now except from the finitely many roots α withdα(y, z) > 0, the inequality 0 ≤ 〈x, α〉 implies 0 ≤ 〈y, α〉, 〈z, α〉, whence

〈x, α〉 ∈ [0, N ] ⇒ 〈y, α〉, 〈z, α〉 ∈ [0, N ].

Thus, the general case follows from the special one by applying it to y. �

5.7.2 Theorem. Let α ∈ Φ, and let A ⊂ W be the corresponding half-space. Then the following are equivalent.

48 Daan Krammer

(1) SpanwZα is positive definite and∑k

n=1wnα = 0, where k is the

smallest positive integer with wkα = α.(2)

⋂n∈Z w

nA =⋂

n∈Z wn(W −A) = ∅, and A is periodic.

(3) Q(w) ⊂ µ(α).(4) Q2(w) ⊂ µ(α).(5) Q1(w) ⊂ µ(α).

As to (1), note that if SpanwZα is positive definite, then α is periodic,by 1.2.6.

Proof. (1)⇒(2). Suppose⋂

n∈ZwnA 6= ∅. Then there exists x ∈ U0 such

that 〈x,wnα〉 > 0 for all n ∈ Z. Hence 0 = 〈x, 0〉 =∑k

n=1〈x,wnα〉 > 0, acontradiction. Hence

⋂n∈Zw

nA = ∅ and similarly⋂

n∈Zwn(W −A) = ∅.

Clearly, A is periodic.(2)⇒(3). We will prove that for periodic A,

⋂n∈Zw

nA = ∅ impliesQ(w) ⊂ K(W−A). Consequently, (2) implies Q(w) ⊂ K(A) ∩K(W−A) =µ(A). Let x ∈ Q(w) ∩ H(A). Let k > 0 be such that wkA = A. ThenwkZx ⊂ H(A). By 5.5.4, we find wZx ⊂ H(A). Let g ∈ W be such thatx ∈ gC. It follows that g ∈ ⋂

n∈ZwnA, a contradiction.

(3)⇒(4)⇒(5) is trivial.(5)⇒(1). By 5.6.10, Q1(w) is non-empty; choose any x ∈ Q1(w). Since

Q1(w) is w-invariant, (5) gives wZx ⊂ µ(α), whence x ∈ ⋂n∈Zw

nµ(α).Since x ∈ U0, it follows that P := SpanwZα is positive definite (for P iscontained in the span of the root system {β ∈ Φ | 〈x, β〉 = 0}, which isfinite by 2.2.5). In particular, as noted above, α is periodic, say wkα = α,k > 0 minimal. Let u =

∑kn=1w

nα, and let u∗ = (u, ·) ∈ E∗. Considery = x+ εu∗, where ε > 0 will be chosen later on.

First note that, by 1.2.2(d), there exists η > 0 such that if 0 < ε < ηthen y ∈ U0, and for all β ∈ Φ: dβ(x, y) > 0 ⇒ x ∈ µ(β). By 5.7.1, thereexists δ > 0 such that for all β ∈ Φ, either 〈x, β〉 = 0 or |〈x, β〉| ≥ δ. LetN > max{|〈u∗, β〉| | β ∈ Φ, x ∈ µ(β)} (note that this is possible sincethere exist only finitely many β ∈ Φ with x ∈ µ(β)). Choose ε such that0 < ε < min(η, δ/N).

We will show that y ∈ Q1(w). It is enough to show that y ∈ Q(w),because the additional properties for Q1(w) follow from the facts x ∈ Q1(w)and wu∗ = u∗. We must show that for all β ∈ Φ, the function Z →{−1/2, 0, 1/2}, n 7→ fβ(wny) is monotonic — see 5.5.4.

If dβ(wnx,wny) = 0 for all n, there is nothing to prove. Otherwise, wemay translate β over a power of w so as to have dβ(x, y) > 0 (and hencex ∈ µ(β) and |〈u∗, β〉| < N by the above). Since wu∗ = u∗, we have for alln ∈ Z:

〈wny, β〉 = 〈wn(x+ εu∗), β〉 = 〈wnx, β〉 + ε〈u∗, β〉.The sign of 〈wny, β〉 is determined by the first of the two terms on theright. More precisely, if 〈wnx, β〉 6= 0 then

|ε〈u∗, β〉| ≤ | δN

〈u∗, β〉| ≤ δ < |〈wnx, β〉|.

In other words, fβ(wnx) 6= 0 ⇒ fβ(wny) = fβ(wnx). Furthermore, if onthe contrary fβ(wnx) = 0, then fβ(wny) has the same sign as 〈u∗, β〉, thatis, a constant independent of n. It follows that n 7→ fβ(wny) is monotonic.Hence y ∈ Q1(w).


Figure 8: Classes of roots

root ��

BBBBBBB

even

periodic

odd

��

@@

��HH

��HH

α or −α supporting (and periodic)

rest

critical

supporting (and even)

non-supporting (and even)

outward

inward

essential

Since x, y ∈ Q1(w) ⊂ µ(α), we have 〈x, α〉 = 〈y, α〉 = 0. It follows that〈u∗, α〉 = 0 or, equivalently, (u, α) = 0. Since wu = u and P = SpanwZα,we have u ⊥ P . Since u ∈ P and P is non-degenerate, we have u = 0. Thisproves (1). �

5.7.3 Definition. A half-space, root or reflection is called w-critical if itsatisfies the equivalent conditions of 5.7.2. It is called w-essential if it iseither w-critical or w-odd.

5.7.4 Examples. (a). Let W be finite. After conjugation, we may writePc(w) = WI . Now Q(w) = KI and the set of critical roots equals ΦI .

(b). Let w be torsion. Then Pc(w) is a finite parabolic subgroup.After conjugating w, we may write Pc(w) = WI . Now Q(w) = {x ∈ U0 |dΦ(x,wx) = 0} = KI . Moreover, α is critical if and only if Q(w) ⊂ µ(α),that is, rα ∈ (KI)

′. Since I is spherical, it is facial, by 2.2.4. By 2.1.4,(KI)

′ = WI . Thus, the set of critical roots equals ΦI .

(c). Let us go back to example (c) of 5.2.5. Using criterion 5.7.2(2), wefind that the only critical wall is the thick line. In 5.5.2(c), we remarked thatQ(w) equals the thick line. The reader is invited to check that conditions5.7.2 (1) · · · (3) are equivalent for all roots.

An overview of our classification of roots is given in Figure 8. A periodicroot α is called w-rest if it is not critical and α and −α are not supporting.An example of a w-rest root is every root, if w = 1. Examples of supportingperiodic roots can be found in example (c) of 5.2.5. It is easily seen that toeach horizontal line different from the thick one, one of the associated rootsis supporting periodic. Another nice example is that (in the same example)every w-periodic root is w2-rest (since Q(w2) = U0 as w2 is a translation— see 5.5.2(d)).

5.7.5 Proposition. Let A ⊂ W be a half-space. Then the following areequivalent.

(1) A is essential.(2)

⋂n∈Z w

nA =⋂

n∈Z wn(W −A) = ∅.

50 Daan Krammer

Proof. (1)⇒(2) follows directly from the definitions of critical and odd.(2)⇒(1). By condition 5.7.2(2), we need only show that A is not even.Suppose it is, say supporting. Let x ∈ Q(w), and let g ∈ W be such thatx ∈ gC. Now from wZx ⊂ Q(w) ⊂ H(A) we find wZg ⊂ A, contradict-ing (2). �

5.7.6 Proposition. The set of critical roots is a finite, parabolic rootsubsystem of Φ.

Proof. This follows immediately from criterion 5.7.2(3), 2.1.4 and the factthat ∅ 6= Q(w) ⊂ U0. �

We know that for an even α, either α or −α is supporting, and odd rootsare never supporting. The following proposition characterizes supportingperiodic roots.

5.7.7 Proposition. Let α be periodic. Then the following are equivalent.

(1) Q(w) ∩ µ(α) 6= ∅.(2) SpanwZα is positive definite.(3) Neither α nor −α is supporting.

Proof. Let f ∈ P [X] be such that f(w)U0 ⊂ Q2(w) (see 5.6.5).

(1)⇒(2). Let x ∈ Q(w)∩µ(α). Let k > 0 be such that wkα = α. ThenwkZx ⊂ µ(α). Since x ∈ Q(w), this implies wZx ⊂ µ(α) by 5.5.4. HenceSpanwZα is positive definite by 2.2.5 and 1.2.6.

(2)⇒(3). By 3.2.3, there exists x ∈ U0 such that x ∈ wnµ(α) for alln ∈ Z. Now f(w)x ∈ Q2(w) ∩ µ(α), whence (3).

(3)⇒(1). Suppose that (3) holds but not (1). Then there are x ∈Q(w)∩H(α) and y ∈ Q(w)∩H(−α). Since x ∈ Q(w) and α is not odd, wehave wZx ⊂ H(α) by 5.5.5. Hence f(w)x ∈ Q2(w) ∩ H(α), and similarlyf(w)y ∈ Q2(w)∩H(−α). Let t > 0 be such that f(w)(x+ty) ∈ µ(α). SinceQ2(w) is convex, we have f(w)(x+ ty) ∈ Q2(w) ∩ µ(α) ⊂ Q(w) ∩ µ(α). �

We finish the subsection with a conjecture on Q(w).

5.7.8 Definition. Define Q∗(x) to be the intersection of all H(α) containingQ(w), and all µ(α) containing Q(w).

5.7.9 Conjecture. Q(w) = Q∗(w). �

The reader is invited to check this conjecture for the examples of 5.5.2(e).Conjecture 5.7.9 does not seem to be more difficult than its special casewhere there are no periodic roots at all.

Conjecture 5.7.9 easily implies the following statements. None of thesestatements however has been proved yet without use of 5.7.9. We note that5.7.10(d) can be reformulated in a purely combinatorial way.

5.7.10 Conjecture.

(a) Q(w) is combinatorially convex, that is, it is the intersection ofa family of (closed or open) half-spaces in U0.

(b) Q(w) is open in the intersection of those µ(α) containing Q(w).(c) Q(w) is convex.


(d) Q(w) is topologically connected. �

5.8 The parabolic closure

For this subsection, fix w ∈W .

Let us call a point x ∈ Q(w) general if for all roots α, the conditionx ∈ µ(α) implies Q(w) ⊂ µ(α), that is, α is critical.

5.8.1 Theorem. Consider the following subgroups of W .

H = the parabolic closure of w,K = the standard parabolic closure of w,L = the subgroup generated by the essential reflections.

If there exists general x ∈ Q(w) such that x ∈ C, then H = K = L.

Proof. H ⊂ K follows from the fact that every standard parabolic sub-group is a parabolic subgroup.

K ⊂ L. It is enough to show that every root separating 1 from w isessential. Let α be a root separating 1 from w. We have

0 < dα(C,wC) ≤ dα(C, x) + dα(x,wx) + dα(wx,wC).

Thus, one of the terms on the right is positive. If dα(x,wx) > 0, it followsthat α is odd, by 5.5.5. If dα(C, x) > 0 then x ∈ µ(α), whence α is critical.Similarly if dα(wx,wC) > 0.

L ⊂ H. Note that H and L, contrary to K, are conjugacy covariant,that is, Pc(gwg−1) = gPc(w) g−1 and similarly for L. By conjugating w,we may suppose H to be standard parabolic. Note that we then no longermay assume x ∈ C ∩Q(w), but we do not need this. Let A be an essentialhalf-space. Then by 5.7.5, wZ 6⊂ A and wZ 6⊂ W − A. Hence H 6⊂ A andH 6⊂ W − A. Let p ∈ H ∩ A, q ∈ H ∩ (W − A). Since H is standardparabolic (that is, connected in the Cayley graph), there exists a path in H(viewed as subgraph of the Cayley graph) from p to q. One of the edges ofthis path is from A to W −A, say {y, ys}. It follows that rA = ysy−1 ∈ H,which proves L ⊂ H. �

We note that in 5.8.1, instead of x being general, it is sufficient that xlies on only essential walls. The proof is the same.

5.8.2 Lemma. Q(w) contains general points.

Proof. By 5.6.2, Q2(w) is a non-empty cone. By 2.1.2, there exists x ∈Q2(w) such that x′ = Q2(w)′. Now for every root α, if x ∈ µ(α) thenQ2(w) ⊂ µ(α), whence α is critical, by 5.7.2. Furthermore, x ∈ Q2(w) ⊂Q(w). Hence x is a general point in Q(w). �

5.8.3 Theorem. The parabolic closure of w equals the subgroup of Wgenerated by the w-essential reflections.

Proof. By 5.8.2, there exists a general point x ∈ Q(w). By conjugating w,we may suppose x ∈ C. The result now follows from 5.8.1. �

Subsequently, theorem 5.8.3 was used in [44].

52 Daan Krammer

5.8.4 Definition. Denote the root system generated by the odd roots byΦodd.

5.8.5 Lemma. Let α be periodic. Then either α ⊥ Φodd or α ∈ Φodd.

Proof. Suppose α is not perpendicular to Φodd, say (α, β) 6= 0, β odd. Wemay suppose β to be outward, and (α, β) < 0. Let γ = rβ α = α−2 (α, β)β.We will show that γ is odd. Let x ∈ U0 be any point. By 5.7.1, we have

limn→∞

〈x,wnβ〉 = ∞, limn→−∞

〈x,wnβ〉 = −∞.

Hence

〈x,wnγ〉 = 〈x,wnα− 2 (α, β)wnβ〉 → ∞, n→ ∞

and similarly 〈x,wnγ〉 → −∞ as n → −∞. Thus, γ is odd. Hence α =rβ γ ∈ Φodd. �

5.8.6 Definition. We define Pc∞(w) to be the group generated by the w-odd reflections, and Pc0(w) to be the group generated by rα for all criticalroots perpendicular to Φodd.

5.8.7 Corollary. We have Pc(w) = Pc∞(w) × Pc0(w). Furthermore,Pc∞(w) is the product of the infinite components of Pc(w).

Proof. The first statement follows immediately from 5.8.5 and 5.8.3. Asto the second statement, for every component H ⊂ Pc(w), at least one ofthe following is true.

(1) H is generated by the odd reflections in H.(2) H is generated by the critical reflections in H.

In case (1) holds, H is infinite, since for every odd reflection r ∈ H, allwnrw−n are different (n ∈ Z). In case (2), H is finite, by 5.7.6. (Inparticular, (1) and (2) cannot both hold.) �

5.8.8 Example. An example of a critical root α ∈ Φodd is the (up to signunique) critical root in the Coxeter group of 5.2.5(c). Namely, Φodd = Φ.

Examples of critical roots α ⊥ Φodd are easily constructed, for examplewith w torsion, that is, Φodd = ∅.

The preceding results do not imply a fast and easy to implement algo-rithm computing Pc(w) for given w. To this end, we designed the followingfunny variation of 5.8.1. Namely, we replace Q(w) by Q∗(w), which isconjecturally the same set, by 5.7.9. Theorem 5.8.1 is an ingredient to theproof of 5.8.9. Again, call a point in Q∗(w) general if it lies on only criticalwalls.

5.8.9 Corollary. Let x ∈ Q∗(w) be a general point, and assume thatx ∈ C. Then Pc(w) is a standard parabolic subgroup of W .

Proof. Let I ⊂ S be such that x ∈ CI . Since x is general, the criticalreflections generate the standard parabolic group WI . The proof will befinished by showing that Pc∞(w) is a standard parabolic subgroup too.By 5.6.3 and 5.3.2, there exists n > 0 such that all w-periodic roots arewn-invariant, and for every outward root α, we have α < wnα. For every


supporting even root α, we have x ∈ Q∗(w) ⊂ H(α) and hence C ⊂ H(α).Now it is easy to see that C ⊂ Q(wn). By 5.8.1, Pc(wn) is a standardparabolic subgroup. Since all periodic roots are wn-invariant, there are nown-critical roots — use, for example, criterion 5.7.2(1). Hence Pc(wn) =Pc∞(wn) = Pc∞(w) by 5.2.7. �

5.8.10 Remark. The above result can be strengthened by replacing Q∗(w)by Q+(w), which is defined to be the intersection of H(α) for all supportingeven α, and µ(α) for all critical α. The difference with Q∗(w) is thatsupporting periodic roots are removed. The proof is the same. It is anopen question whether Q+(w) = Q∗(w). It is true for the examples of5.5.2(e).

5.9 Calculations in number fields

For the subsequent subsection with algorithms on Coxeter groups, we needsome facts on algorithms for calculations in number fields and their com-plexity. These algorithms are guaranteed to give the correct answers. Theycompute with algebraic numbers without rounding. Therefore, the algo-rithms cannot be compared to numerical algorithms doing similar things.

LetK be a number field, that is, a finite extension of Q. Given a Q-basisxi of K, we define the height on K by

h(∑

i

aixi/a) = log(∑

i

a2i + a2),

where ai and a are relatively prime integers. The height of a polynomial (orrational function, or matrix) is by definition the maximum of the heightsof its coefficients (or entries).

5.9.1 Calculations in number fields. Let x, y ∈ K. Then x + y can becomputed in O(h(x) + h(y)) time, whereas xy and x/y can be computedin O(h(x)h(y)) time. There exists a faster algorithm with complexityO((h(x) + h(y))1+ε) for every ε > 0, using Fourier transforms [39, p. 278].We will not consider this improvement of the bounds of complexity.

5.9.2 Signs of real algebraic numbers. Fix a real number field K. Thenthere exists an algorithm that, given f, g ∈ K[X] and an integer i, decideswhether g(a) < 0 or g(a) = 0 or g(a) > 0, where a is the i-th real root off in the natural ordering (supposing a exists). Its complexity is O(n6h2),where n is a bound for the degrees of f, g, and h a bound for their heights[21, p. 93]. See also [29], [24, section 3.2.1.1]. It is easy to deduce fromTarski’s generalization of Sturm’s theorem [38, VI.10] that for fixed n, thecomplexity is O(h2). This suffices for us, since we are only interested in thecase where n is bounded (because we fix one Coxeter group). A particularcase is that for all x ∈ K, it can be decided in O(h(x)2) time whetherx > 0.

5.10 Algorithms

In this subsection, we give a chain of algorithms, leading to a computa-tion of the parabolic closure. The algorithms make extensive use of therepresentation and the Tits cone.

54 Daan Krammer

Fix a root basis (E, (·, ·),Π) defined over a real number field K — see4.7.5. For each of the algorithms, part of the input is w ∈ W , given by astring. The first step in every algorithm is the computation of the matrixof w and, in some cases, the roots separating 1 from w. By 5.9.1, this canbe done in O(ℓ(w)2) time. The height h(w) is O(ℓ(w)).

Following [10], define the depth of a root α to be

dp(α) =ℓ(rα) + 1

2.

We have h(α) = O(dp(α)). For a vector x ∈ U0, defined over a (fixed)number field, the quantity max{h(x), dΦ(x,C)} will also be called depthand denoted by dp(x) — we are running out of names, and the two notionsof depth will be used for the same purpose.

An algorithm is called quadratic, cubic, . . . if has complexity O(ℓ(w)2),O(ℓ(w)3), etc.

5.10.1 Proposition. There exists an algorithm that, given a root α, de-cides whether it is periodic, critical, odd, even. Moreover, for odd rootsit decides whether it is outward or inward, and for even roots it decideswhether it is supporting or not. If dp(α) = O(ℓ(w)) then the algorithm isquadratic.

Proof. It is easy to decide whether α is periodic. Using condition 5.7.2(1),it is easy to decide whether α is critical. Both are quadratic in time, by5.9.1 and 5.9.2. Let us suppose α to be non-periodic. We will describethe algorithm deciding whether α is odd or even. First find n > 0 suchthat |(α,wnα)| ≥ 1, by simply trying n = 1, 2, . . . By 5.3.2, such an nexists and is bounded. Moreover, by 5.3.2 again, α is odd if (α,wnα) ≥ 1and α is even if (α,wnα) ≤ −1. This enables us to distinguish betweenodd and even. Now suppose α to be odd. By 1.4.7, we have α < wnα orwnα < α. By 1.4.7(d), for all different roots β, γ, we have β < γ if andonly if 〈U0, γ − β〉 ⊂ R>0. Hence, choosing any point u ∈ U0 (for example,u ∈ C), we have

α is outward ⇔ α < wnα⇔ 〈u,wnα− α〉 > 0.

This provides a simple test whether α is outward or inward. A similarargument shows that an even root α is supporting if and only if 〈u, α +wnα〉 > 0. �

5.10.2 Definition. Let us call an outward half-space A ⊂ W canonical if1 ∈ A but wn 6∈ A for all n = 1, 2, . . . Similarly for roots.

Note that, for every root α, the orbit wZα contains a unique canoni-cal root. Contrary to our classifications of roots so far, canonicity is notconjugacy covariant.

5.10.3 Proposition. There exists an algorithm that, given an outwardroot, decides whether it is canonical or not. The algorithm is quadratic,provided the root has linear depth.

Proof. Let α be outward. Compute n > 0 such that (α,wnα) ≥ 1. Then


A ⊂ wnA, where A = A(α). It follows that A is canonical if and only if1 ∈ A, and for all k = 1, 2, . . . , n, we have wk 6∈ A. This is easily tested. �

5.10.4 Proposition. The algorithm of 1.2.4 is cubic in dp(x). Thereexists an algorithm computing dΦ(x, y) which is cubic in dp(x) + dp(y).

Proof. As to the algorithm of 1.2.4, computing the next xk takes quadratictime in dp(x), and this has to be done about dΦ(x,C) times. The otherstatement follows easily. �

5.10.5 Proposition. There exists a cubic algorithm that finds all criticalroots (up to wZ), all canonical outward roots, and r(w). There existsan algorithm testing whether a given point in U0 is in Q(w). The latteralgorithm is cubic if the point has linear depth.

Proof. The canonical outward roots can be found by computing the half-spaces A with 1 ∈ A, w 6∈ A, and testing them on outwardness (5.10.1)and canonicity (5.10.3). Since here, a quadratic test must be applied ℓ(w)times, this is a cubic algorithm. Now r(w) is simply equal to the numberof canonical outward roots. By 5.5.7, x ∈ Q(w) if and only if dΦ(x,wx) =r(w), which can easily be tested in cubic time by 5.10.4. By 5.7.5, for everycritical root α, there exists n ∈ Z such that wnα separates 1 from w. Thus,the critical roots are found (up to wZ) by testing all roots separating 1 fromw on criticality. �

5.10.6 Proposition. There exists a cubic algorithm that computes a pointz ∈ Q(w) with dp(z) = O(ℓ(w)).

Proof. The algorithm is as follows. Choose any point x ∈ U0. Computethe projection y of x on the intersection of the critical walls. (This is notnecessary but may speed up the algorithm.) For n = 0, 1, 2, . . ., try whetherthe point zn = (1 + w + · · · + wn) y is in Q(w). By 5.6.5, some zn is inQ(w), with bounded n. Hence the estimate of dp(z). Testing whether onezn ∈ Q(w) is cubic in time, by 5.10.5. Since n is bounded, the completealgorithm is cubic. �

5.10.7 Lemma. Let x ∈ U0. Then x ∈ Q∗(w) if and only if x is on everycritical wall, and there is no supporting root separating x from wx.

Proof. ‘If’. Let α be supporting. We must show x ∈ H(α). If not,we have wZx ⊂ K(−α), since wnx and wn+1x are not separated by asupporting root. Let f ∈ P [X] be such that f(w)U0 ⊂ Q(w), see 5.6.5.Then f(w)x ∈ K(−α) ∩ Q(w) = ∅, a contradiction. Hence x ∈ H(α),which proves ‘if’. The converse implication is trivial. �

5.10.8 Proposition. There exists an algorithm that tests whether a givenpoint is in Q∗(w). The algorithm is cubic if the point has linear depth.

Proof. Lemma 5.10.7 suggests the following algorithm. Let x ∈ U0 begiven. By conjugating w, we may suppose x ∈ C (in cubic time if dp(x) =O(ℓ(w)), by 5.10.4). Compute the roots separating x from wx. Decidewhether one of them is supporting by 5.10.1, and (2)⇔(3) of 5.7.7. This is

56 Daan Krammer

also cubic if x has linear depth. �

5.10.9 Proposition. There exists a cubic algorithm that computes I ⊂ S,g ∈W such that Pc(w) = gWI g

−1.

Proof. The algorithm is as follows. Compute a facet X in Q(w). Next, wecompute a general facet Y ⊂ Q∗(w) (that is, the points in Y are general),as follows. If X is not general, then, since Q∗(w) is open in the intersectionof the critical walls, there exists a facet Y ⊂ Q∗(w) such that X ⊂ Y anddim Y = dim X + 1. Such a facet Y can be found using 5.10.8. Going onthis way, we end up with a general facet Y ⊂ Q∗(w). By conjugating w,we may suppose Y ⊂ C. By 5.8.9, Pc(w) equals the standard parabolicclosure of w, which can be computed using 1.3.6. �

In the algorithm of 5.10.9, a general point in Q∗(w) is computed. Con-jecturally (5.7.9), it is a general point in Q(w). Another way of computinga general point in Q(w) is suggested by the proof of 5.8.2. However, thisalgorithm is not as easy to implement as the algorithm of 5.10.9. This iswhy we prefer to use 5.8.9.

6 The conjugacy problem

In 3.1.13 and 5.10.9, the conjugacy problem for Coxeter groups has beenreduced to the case Pc(v) = Pc(w) = W . In this section, we solve theconjugacy problem for this case.

We will treat affine and non-affine groups separately. The affine groupshave been taken care of in 4.2.1. In subsection 6.1, it is shown that forirreducible infinite non-affine groups, there exists a non-degenerate rootbasis. In subsection 6.3 we assume in addition that Pc(w) = W , and provesome crucial results, for example, for all outward α, β one has α < wnβ foralmost all n > 0 (6.3.9). In subsection 6.4, we give our first polynomialsolution to the conjugacy problem, with exponent #S+3. In subsection 6.5,it is shown, amongst others, that under the above conditions, the maximumof the absolute values of the eigenvalues of w is a simple eigenvalue (6.5.14).This opens the way to a cubic solution to the conjugacy problem (6.7.4). Insubsection 6.8, the results of subsection 6.3 are applied to the computationof the algebraic rank of a Coxeter group.

6.1 Dividing out the radical

6.1.1 Lemma. Let (W,S) be an irreducible Coxeter system, associated toa root basis (E, (·, ·),Π). Suppose there exists x =

∑α∈Π λαα 6= 0 in the

radical E⊥ of E, such that λα ≥ 0 for all α ∈ Π. Then (W,S) is affine.

Proof. We may suppose Π to be a basis of E. Namely, if Π is not a basisof E, replace E by an abstract vector space with basis Π and a symmetricbilinear form (·, ·)′ such that (α, β)′ = (α, β) for all α, β ∈ Π. Let A = (ast)be the Gram matrix. Let (k, ℓ,m) be the signature of E. Note that m ≥ 1.Suppose that ℓ + m > 1. Then A has an eigenvector y, independent ofx, with eigenvalue λ ≤ 0. Consider the non-negative irreducible matrixB = 1 − A. By 1.5.1, since x is a non-negative eigenvector of B, the


eigenvalue of B at x is greater than at y, contradicting λ ≤ 0. Henceℓ+m = 1, which implies that (W,S) is affine. �

Let (E, (·, ·),Π) be a root basis. By dividing out E⊥, we obtain a triple(E/E⊥, (·, ·),Π′). We ask ourselves whether the new triple is still a rootbasis. Of the axioms for a root basis 1.2.1, parts (1) and (2) certainly hold.

6.1.2 Proposition. Let (W,S) be an irreducible non-affine Coxeter sys-tem. If the triple (E, (·, ·),Π) is associated with (W,S) and satisfies (1) and(2) of definition 1.2.1 then it satisfies (3) as well.

Proof. Suppose that (3) is not satisfied. Then there are λα ≥ 0 (α ∈ Π),not all zero, such that

∑α∈Π λαα = 0. There exists a vector space E′ with

a basis Π′ and a bijective map φ: Π′ → Π. Let L: E′ → E be the uniquelinear map such that L(α) = φ(α) for all α ∈ Π′. For u, v ∈ E′ define(u, v) := (Lu,Lv). The element

∑α∈Π λα φ

−1(α) is in the radical of E′.This contradicts 6.1.1 and finishes the proof. �

6.1.3 Proposition. Let (E, (·, ·),Π) be a root basis, associated to an irre-ducible non-affine Coxeter system (W,S). Then by dividing out the radicalE⊥, one obtains again a root basis associated to (W,S).

Proof. Immediate from 6.1.2. �

Proposition 6.1.3 is implicit in the work of Vinberg, [47, Prop. 13,p. 1100].

6.1.4 Example. By 6.1.3, every irreducible non-affine Coxeter group is as-sociated to a non-degenerate root basis (E, (·, ·),Π). We cannot suppose Πto be independent, as is shown by the following example. Let S = Z/n,mst = 2 if s− t 6= ±1, mst = 4 if s− t = ±1. Then

∑s∈S ases ∈ E⊥ if and

only if for all s ∈ S, as − (as−1 + as+1)/√

2 = 0, or, equivalently,

(as

as+1

)=

(0 1

−1√

2

)(as−1

as

).

Since the above 2 × 2 matrix has order 8, the classical root basis is degen-erate if n is an 8-tuple. Suppose n is an 8-tuple. Then the radical of theclassical root basis has dimension 2. Since all mst are finite, every rootbasis (spanned by Π) is a quotient of the classical root basis. Hence inevery non-degenerate root basis, Π is dependent.

6.2 Ordered triples of roots

6.2.1 Lemma. A root basis (E, (·, ·),Π) with E = Span(Π) cannot havesignature (1, 2) or (1, 1, 1).

Proof. Suppose dim E = 3. Let {α, β, γ} ⊂ Π be a basis of E. Let A bethe Gram matrix with respect to this basis. Thus,

A =

1 −a −c−a 1 −b−c −b 1

, a, b, c ≥ 0.

58 Daan Krammer

If one of a, b, c is smaller than 1, say a < 1, then there is a subspaceSpan {α, β} of signature (2, 0), which makes signatures (1, 2) and (1, 1, 1)impossible. Otherwise, since a, b, c ≥ 1, we have

det(A) = 1 − 2 abc− (a2 + b2 + c2) ≤ 1 − 2 − 3 < 0,

which again makes signatures (1, 2) and (1, 1, 1) impossible. �

6.2.2 Proposition. Let α ≤ β ≤ γ be roots. Let us write 2 (α, β) =p+ p−1, 2 (β, γ) = q + q−1, 2 (α, γ) = r + r−1, p, q, r ≥ 1. Then r ≥ pq.

Proof. By 1.4.7(e), we may suppose Φ to be generated by {α, β, γ}. As-sume also that E = Span Φ. Then dim E ≤ 3. If E is positive definiteor semi-definite, then for all x, y ∈ E with (x, x) = (y, y) = 1 we have|(x, y)| ≤ 1. Hence p = q = r = 1, and the result follows. Thus, supposeE is not positive definite nor semi-definite. By 6.2.1 (and noting that Econtains a positive definite subspace Rα), E has signature (2, 1) or (1, 1).

Since E is non-degenerate, it may be identified to its dual. Let H ={x ∈ E | (x, x) < 0}. Let H+, H− be the two connected components of H.By 4.6.1, U0 contains either H+ or H−.

There are two ways of finishing the proof. The first way uses the geom-etry of the hyperbolic plane, as follows. Suppose E to have signature (2, 1),and suppose p, q, r > 1. Since U0 contains H− or H−, α, β and γ definethree parallel geodesics in H2, ordered in this order. Let us denote thesegeodesics by α⊥, β⊥, γ⊥. It is known [5, Cor. A.5.8] that the hyperbolicdistance between α⊥ and β⊥ is log(p), and similarly for the other pairs ofgeodesics. The (unique) shortest path from α⊥ to γ⊥ intersects β⊥. Thisshows log(r) ≥ log(p) + log(q), whence r ≥ pq. The cases where E hassignature (1, 1), or one of p, q, r equals 1, are left to the reader in our firstend of the proof.

The second way of finishing the proof will not depend on any knowledgeof the hyperbolic plane.

Write a = (α, β), b = (β, γ), c = (α, γ). Let A be the Gram matrixwith respect to α, β, γ:

A =

1 a ca 1 bc b 1

, a, b, c ≥ 0.

We have

0 ≤ −det(A) = c2 − 2 abc+ (a2 + b2 − 1),

whence

c ≤ ab−√

(a2 − 1)(b2 − 1) or c ≥ ab+√

(a2 − 1)(b2 − 1). (9)

Note that we may assume a > 1 or b > 1, for otherwise, we have p = q = 1and certainly r ≥ pq. By symmetry, we may assume a > 1.

We saw that U0 contains either H+ or H−. Combining with 1.4.7(d),we find that for all x ∈ H, the sequence (x, α), (x, β), (x, γ) is monotonic.We apply this to x = α − β. We have (x, x) = 2 − 2 a < 0, whence indeedx ∈ H. By the above, the sequence ((x, α), (x, β), (x, γ)) = (1−a, a−1, c−b)


is monotonic. Since it was assumed that a > 1, we find it to be increasing,and hence a− 1 ≤ c− b, or

c ≥ a+ b− 1. (10)

If b = 1, then from (10), we find c ≥ a, which gives the desired result.Therefore, let us assume a, b > 1. We will show that (10) contradicts thefirst possibility of (9). These two inequalities would imply successively

a+ b− 1 ≤ c ≤ ab−√

(a2 − 1)(b2 − 1),√

(a2 − 1)(b2 − 1) ≤ ab− (a+ b− 1),

(a2 − 1)(b2 − 1) ≤ (a− 1)2(b− 1)2,

and as a, b > 1,(a+ 1)(b+ 1) ≤ (a− 1)(b− 1),

a contradiction indeed. Hence the second possibility of (9) holds. In termsof p, q, r one finds

c ≥(p+ p−1)(q + q−1) +

√((p+ p−1)2 − 4)((q + q−1)2 − 4)

4=

=(p+ p−1)(q + q−1) + (p− p−1)(q − q−1)

4=pq + (pq)−1

2.

Hence r ≥ pq. �

6.2.3 Corollary. Let α ≤ β ≤ γ be roots. Then

(α, β) ≤ (α, γ)

and(β, γ) ≤ (α, γ).

Proof. This follows from 6.2.2 and the fact the map x 7→ x + x−1 isincreasing for x ≥ 1. �

6.3 Outward roots and their ordering

In this subsection, we assume (W,S) to be irreducible, infinite and non-affine, unless stated otherwise. Furthermore, an element w ∈ W withPc(w) = W is fixed. By 5.8.7, we have Pc∞(w) = W , and Span(Π) isspanned by the outward roots.

By 6.1.3 there exists a non-degenerate root basis (E, (·, ·),Π) for W .Also we may assume that E is spanned by Π and hence by the odd roots.Such E is fixed from now on.

One of the main results of this subsection is corollary 6.3.8 stating thatfor all outward α, β one has (α,wnβ) → ∞ as n → ∞ or n → −∞. Aresult of independent interest is corollary 6.3.10 stating that wZ has finiteindex in the centralizer of w.

6.3.1 Lemma. Let α, β be w-outward. Then(a) (α, β) > −1.(b) For almost all n ∈ N, (α,wnβ) ≥ 1 implies α < wnβ.

60 Daan Krammer

Proof. Part (a) is left to the reader (use 1.4.7(b) and the definition ofoutwardness). In order to prove (b), let x ∈ U0 be any point. By def-inition of outwardness, 〈x,wnβ〉 > 0 for almost all n ∈ N. By 5.7.1,limn→∞〈x,wnβ〉 = ∞. Hence for almost all n ∈ N, 〈x,wnβ − α〉 > 0.For these n, the additional condition (α,wnβ) ≥ 1 implies, by 1.4.7(b),that α < wnβ. �

6.3.2 Lemma. Let α, β be outward roots such that (β,wnα) is unbounded(n ∈ N). Then there exists k ∈ N such that w−kα < β.

Proof. Recall that (Φ,Φ)∩ (−1, 1) is finite by 3.2.1(a). For infinitely manyk ∈ N we have |(w−kα, β)| ≥ 1 because otherwise the sequence (wnα, β)would be periodic, contradicting its unboundedness. By 6.3.1(a) there isan infinite set L ⊂ N such that (w−kα, β) ≥ 1 for all k ∈ L. By 6.3.1(b)almost all k ∈ L satisfy α < wkβ. �

6.3.3 Lemma. Let α be odd. Then (α,wnα) is unbounded (n ∈ N).

Proof. Since W is irreducible and non-affine, we may assume that E isnon-degenerate, by 6.1.3. Also, we may assume that E is spanned by Πand hence by the odd roots.

The set wNα is a discrete subset of E by 1.2.5 and it is infinite since αis odd. Hence it is unbounded. Since the odd roots span E, and E is non-degenerate, there exists an odd root β such that (β,wnα) is unbounded(n ∈ N). Suppose, as we may, that α and β are outward. By 6.3.2, wehave w−kα < β for some k ∈ N. By 6.3.1, there exists M > 0 such thatfor all n with n > M and (β,wnα) ≥ 1, we have β < wnα. In order toshow that (α,wnα) is unbounded, let N ≥ 1. Let n > M be such that(β,wnα) ≥ N — see 6.3.1(a). Thus, w−kα < β < wnα. By 6.2.3, we have(α,wk+nα) = (w−kα,wnα) ≥ (β,wnα) ≥ N . �

The preceding lemma is a crucial step to our results. Notice the simi-larity to 5.3.1. The result of 6.3.3 however, does not hold for affine groups.If the root basis is positive semi-definite, then for all roots α, β, we have|(α, β)| ≤ 1.

6.3.4 Definition. Let us call two odd roots α, β equivalent, notation α ∼β, if (α,wnβ) is unbounded, n ∈ N.

6.3.5 Lemma. Equivalence of odd roots is an equivalence relation.

Proof. Reflexivity. This is the content of 6.3.3.

Symmetry. Let α ∼ β. We may suppose α and β to be outward. LetN > 1. By 6.3.2, applied to α, β interchanged, there exists k ∈ N such thatβ < wkα. By 6.3.3 and 6.3.1(a), there exists n > 0 such that (β,wnβ) ≥ N .Note β < wnβ. Applying 6.2.3 to the sequence β < wnβ < wn+kα shows(β,wn+kα) ≥ (β,wnβ) ≥ N . Hence β ∼ α.

Transitivity. Let α ∼ β and β ∼ γ. We may suppose α, β and γ to beoutward. Let N ≥ 1. By 6.3.1(b), there exist m > n > 0 such that α <wnβ < wmγ, (α,wnβ) ≥ N . By 6.2.3, we have (α,wmγ) ≥ (α,wnβ) ≥ Nand so α ∼ γ. �


The proof of the following theorem uses symmetry as well as transitivityof ∼ (6.3.5).

6.3.6 Theorem. Let α, β be odd. Then α ∼ β if and only if wZα 6⊥ β.

Proof. ‘Only if’ is trivial. In order to prove ‘if’, one may assume (α, β) 6= 0,for otherwise, replace β by some wnβ. Suppose α 6∼ β. By symmetry of theequivalence relation ∼ (6.3.5), the sequence (α,wnβ) is bounded (n ∈ Z).Consider the root

γ = rβ α = α− 2 (α, β)β =: α+ λβ.

We will show that γ is odd. We have

(γ,wnγ) = (α+ λβ,wn(α+ λβ)) = (α,wnα) + λ2(β,wnβ) + bounded.

We have (α,wnα), (β,wnβ) > −1 for all n by 6.3.1(a). Moreover, by 6.3.3,we have lim supn→∞(α,wnα) = ∞. It follows that lim supn→∞(γ,wnγ) =∞, so γ is odd by 5.3.2. Now γ is equivalent to α as well as β, because

(α,wnγ) = (α,wnα) + bounded,

(β,wnγ) = λ(β,wnβ) + bounded

and λ 6= 0. Hence α and β are equivalent, a contradiction. This finishesthe proof. �

6.3.7 Corollary. Two odd roots are always equivalent.

Proof. Suppose that the set of odd roots can be written X ⊔Y , X,Y 6= ∅,X ⊥ Y . Since Φ is generated by the odd roots, it would follow that Φ isreducible, a contradiction. Hence a partition as above is impossible. By6.3.6, two odd roots are always equivalent. �

6.3.8 Corollary. Let α, β be outward. Then (α,wnβ) → ∞ as n→ ∞ orn→ −∞.

Proof. By 6.3.3, there exists m > 0 such that (α,wmα) > 1. So α < wmα.By applying 6.2.2 inductively to the sequences α < wnmα < w(n+1)mα, wefind that (α,wnmα) → ∞ as n → ∞. It is enough to show that for allr ∈ N we have (α,wnm+rβ) → ∞ as n→ ∞ — the case n→ ∞ follows byinterchanging α and β. Fix such an r. Note Pc(wm) = W , since Pc(wm) ⊃Pc∞(wm) = Pc∞(w) = W . By applying 6.3.7 and 6.3.1 to wm instead of wand the wm-outward roots α and wrβ, we find that α < wkm+rβ for somek ∈ Z. By 6.2.3 applied to the sequence α < wnmα < w(n+k)m+rβ, we find(α,w(n+k)m+rβ) ≥ (α,wnmα) → ∞ as n→ ∞. �

6.3.9 Corollary. Let α, β be outward. Then w−nα < β < wnα for almostall n ∈ N.

Proof. This follows from 6.3.8 and 6.3.1. �

6.3.10 Corollary. We have [Z(w) : wZ] <∞.

62 Daan Krammer

Proof. Recall the set Out(w) of outward roots. The group Z(w) permutesthe finite set wZ\Out(w). Let G be the kernel of this action. Thus, [Z(w) :G] < ∞. We will prove G = wZ. Let g ∈ G. Choose any outward root α.By multiplying g by a power of w, we may suppose gα = α. Let β be anoutward root and write gβ = wkβ. We have for all n ∈ Z:

(α,wnβ) = (gα, gwnβ) = (α,wngβ) = (α,wn+kβ).

By 6.3.8 however, (α,wnβ) is unbounded. Thus k = 0, whence gβ = β.Since the outward roots span E, it follows that g = 1. �

The above proof in fact shows that the Z(w)/wZ-action on wZ\Out(w)is faithful.

Corollary 6.3.10 was subsequently used in [15], [16], [18].In the proof of the following theorem, the root basis E/E⊥ is used once

again.

6.3.11 Theorem. Let F denote the set of w-periodic roots. Then F is afinite parabolic root subsystem of Φ.

Proof. As in the proof of 6.3.3, we may assume E to be non-degenerateand spanned by the odd roots. We will prove that for every periodic rootα and odd root β, we have |(α, β)| < 1. Suppose the contrary. We mayassume β to be outward and α < β — see 1.4.7(b). Choose any pointx ∈ A(α). Let k > 0 be such that wkα = α. Let n > 0 be such thatwknx ∈ A(−β). Then wknx ∈ A(−α). Hence x ∈ A(−w−knα) = A(−α),a contradiction. We conclude |(α, β)| < 1. Since E is non-degenerate andthe odd roots span E, the set F is bounded. Since F is discrete by 5.7.1,it is finite.

It is easy to see that F is a root subsystem and that F = Span(F )∩Φ.By 3.2.3, F is a parabolic root subsystem of Φ. �

6.3.12 Corollary. LetW be a Coxeter group. Then there exists a constantN = N(W ) such that for all w ∈ W and every w-periodic root α, we havewNα = α. The constant N may be taken to be the least common multipleof the orders of elements of

NO(EI)(WI)

for all spherical I ⊂ S.

Proof. Let α be w-periodical. By 5.8.5, we have either α ⊥ Φodd orα ∈ Φodd. First, suppose α ⊥ Φodd. Since Pc(w) = Pc∞(w)×Pc0(w) (5.8.7)and Pc0(w) is a finite parabolic subgroup (5.7.6), we have wN ∈ Pc∞(w).Hence wNα = α.

Next, suppose α ∈ Φodd. Let P be the component of Pc∞(w) containingrα. We may suppose P to be standard parabolic, say P = WJ . Let w denotethe projection of w on WJ . Note that WJ is infinite, by 5.8.7. We considerseparately the cases of WJ affine or non-affine.

If WJ is affine, let T ⊂WJ be the translation subgroup. Then WJ/T ∼=WI for some I ⊂ J . Let π: WJ →WI be the projection. Then (πw)N = 1,because

πw ∈WI ⊂ NO(EI)(WI).


Figure 9: Illustration for 6.4.1

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

· · ·

· · ·

αβ

y wy wny wn+1y

x wx

Hence wN is a translation. Since α is w-periodic, we must have wNα = α,whence wNα = α.

Now suppose WJ to be non-affine. Note WJ = Pc(w). Hence, by 6.3.11,the set of w-periodic roots in ΦJ is a finite parabolic root subsystem of ΦJ ,say ΦI , I ⊂ J . Since

w|EI∈ NO(EI)(WI)

we find wNα = α. �

6.4 First solution to the conjugacy problem

In the following results 6.4.1, 6.4.2 and 6.4.3, we suppose W to be irre-ducible, infinite, non-affine.

The following lemma says that two odd walls cannot be too far frombeing parallel in some sense. Recall c := max(2,#{(α, β) | α, β ∈ Φ} ∩(−1, 1)).

6.4.1 Lemma. There exists a constant N = N(W ) such that the followingholds. Let w ∈ W with Pc(w) = W . Let α, β be two w-outward roots,and let x, y ∈ Q(w), n ∈ Z be such that α and β separate x from wx,α separates y from wy and β separates wny from wn+1y, as is shown inFigure 9. (Separation need not be strict.) Then |n| ≤ N . More precisely,N may be taken to be (c+ 2)r + r.

Proof. We may assume n ≥ 0. Now for every k with 1 ≤ k ≤ n − 1, wehave x ∈ K(α) ∩ K(−w−kβ), wy ∈ K(−α) ∩ K(w−kβ). Since α and βare outward, K(α)∩K(w−kβ) and K(−α)∩K(−w−kβ) are non-empty aswell. By 1.4.7(b), it follows that either |(α,w−kβ)| < 1 or α = ±w−kβ.Thus, |(α,w−kβ)| ≤ 1. Write (α,w−kβ) = ak. Consider the r-tuples(am+1, . . . , am+r) where 1 ≤ m + 1, m + r ≤ n − 1. These r-tuples aredifferent because (ak) is not periodic, by 6.3.8. Thus, the number of r-tuples equals n− r ≤ (c+ 2)r, whence n ≤ (c+ 2)r + r. �

The following result roughly says that ‘slices’ of Q(w) have diameterO(ℓ(w)) if Pc(w) = W .

6.4.2 Theorem. There exists a constant N = N(W ) such that the fol-lowing holds. Let w ∈ W be such that Pc(w) = W . Let x, y ∈ Q(w) andsuppose that there exists an odd root separating x from wx and y from wy.Then dΦ(x, y) ≤ N ℓ(w).

64 Daan Krammer

Proof. Let α be an odd root separating x from wx and y from wy. Westart by showing that dΦ(x, y) equals the number of outward roots strictlyseparating x from y plus a bounded number. By 6.3.11, the number ofperiodic roots is bounded by max{#ΦI | I ⊂ S spherical}. There areno even roots separating x from y because if β is even then either H(β) orH(−β) contains Q(w) and hence x and y. Finally, there are only boundedlymany roots non-strictly separating x from y. Thus, one needs only boundthe number of outward roots strictly separating x from y.

Let β be such a root. Let k, ℓ be integers such that β separates wkx fromwk+1x and wℓy from wℓ+1y. Here, separation need not be strict. Applying6.4.1 to α and w−kβ shows

|k − ℓ| ≤ N0 := (c+ 2)r + r. (11)

We will now show (k +

1

2

)(ℓ+

1

2

)< 0. (12)

First suppose k, ℓ ≥ 0. Then x, y ∈ K(β), contradicting the assumptionthat β strictly separates x from y. Similarly, it is disproved that k, ℓ < 0,which proves (12). From (11) and (12), it follows that k and ℓ are bounded,and more precisely −N0 ≤ k ≤ N0 − 1. Thus, β separates w−N0x fromwN0x. We conclude

dΦ(x, y) ≤ 2N0 r(w) + a bounded number = O(ℓ(w)). �

6.4.3 Lemma. There exists an algorithm that, given w ∈W with Pc(w) =W , and w-outward roots α, β separating 1 from w, determines the smallestinteger n such that α ≤ wnβ. If α and β have linear depth, then thecomplexity is quadratic.

Proof. Note n ≥ 0. Thus, one algorithm runs by simply trying n = 0, 1, . . .,using 1.4.7(d) to decide whether α ≤ wnβ. The smallest n satisfying thisis what we look for. By an argument similar to 5.3.2, n is bounded. Foreach n, quadratic time is needed, whence the complexity is quadratic. �

6.4.4 Theorem. Coxeter groups have polynomial time conjugacy prob-lem, with exponent r + 3 = #S + 3.

Proof. The algorithm is as follows. Let v, w ∈ W . By 5.10.9, computethe parabolic closures of v and w. By 3.1.13, it is enough to solve theconjugacy problem for the case Pc(v) = Pc(w) = W . We may suppose Wto be irreducible and infinite. By 4.2.1, we may suppose W to be non-affine.Choose a non-degenerate root basis for W — see 6.1.3. By 5.10.6, computepoints x ∈ Q(w), z ∈ Q(v) of linear depth. By conjugating v and w, wemay suppose x, z ∈ C. Compute the canonical v-odd and w-odd rootsby 5.10.5. Let α1, . . . , αk (k ≤ r) be a basis of E consisting of v-outwardroots, and suppose that α1 separates z from vz. Let β1, . . . , βn (n = r(w))denote the w-outward roots separating x from wx and with x ∈ H(βi). Thealgorithm so far takes cubic time. If some βi separates wm from wm+1 forsome m, then m is bounded. The same may be supposed to hold for theαi. Hence, in O(ℓ(w)2n2) = O(ℓ(w)4) time, we can compute the smallestintegers p(i, j), q(i, j) such that αi ≤ vp(i,j)αj , βi ≤ wq(i,j)βj by 6.4.3.


We postpone the continuation of the algorithm for a theoretical obser-vation. If v and w are conjugate, then there exists g with gvg−1 = w andgα1 = βi for some i, after multiplying g at the left by a suitable powerof w. Let X denote the set of injective maps φ: {vZαi | i = 1, . . . , k} →wZ\Out(w). Each g ∈ W with gvg−1 = w induces an element of X.We will show that g is determined by its image in X, and the conditionthat there exists i with gα1 = βi. Let m, j be such that m > 1 andgvZαm = wZβj . Since p(1,m) is the smallest integer with α1 ≤ vp(1,m)αm,and q(i, j) is the smallest integer with gα1 = βi ≤ wq(i,j)βj , we must havegαm = wq(i,j)−p(1,m)βj . This determines g, because E is spanned by theαm.

Thus, the algorithm may be continued as follows. For each φ ∈ X, dothe following. Compute A ∈ End(E) such that

(1) Aα1 = βi where φ(vZα1) = wZβi.

(2) If φ(vZαm) = wZβj (m > 1) then Aαm = wq(i,j)−p(1,m)βj .

Next, check whether Av = wA. It may also be checked whether A ∈O(E, (·, ·)). This is not necessary but may speed up the algorithm. If oneof these conditions is not fulfilled, A is left from consideration. Now, onemust decide whether or not A ∈ W , where W is viewed as a subgroup ofGL(E). Suppose A ∈ W . Write y = Az ∈ AQ(v) = Q(w). Note thatAα1 separates x from wx and y from wy. By 6.4.2, ℓ(A) ≤ M := N ℓ(w)for some (known) constant N . This shows how to decide whether A ∈ W .Namely, compute A = A0, A1, . . . which are defined by Ai+1 = sAi, wheres ∈ S such that 〈AiC, es〉 ⊂ R<0. Then A ∈ W if and only if for somei ≤M , we have Ai = I.

Now v and w are conjugate if and only if for at least one φ ∈ X, we haveAvA−1 = w and A ∈W . Note #X = n(n−1)(n−2) · · · (n−k+1) ≤ nk =r(w)k ≤ ℓ(w)r. Since checking one A takes cubic time (compare 5.10.4),the algorithm has polynomial complexity of degree max(4, r+3) and hencer + 3 (if 4 > r + 3, then the theorem is trivial). �

6.4.5 Remark. The complexity of the algorithm of 6.4.4 depends (not onlytheoretically but also in practice) on the constant of 6.4.2, namely, in theprocess of deciding whether A ∈ W . This constant is very large. This isan important practical disadvantage of 6.4.4.

6.4.6 Lemma. Let G = Zn ⋊ F be the semi-direct product of Zn and afinite group F acting on Zn. Let ℓ be the length function with respect tosome finite generating subset of G. Then there exists a constant N suchthat if v, w ∈ W are conjugate, then there exists g ∈ W with gvg−1 = wand ℓ(g) ≤ N(ℓ(v) + ℓ(w)).

Proof. The proof goes along the same lines as 4.2.1, and is left to thereader. �

The following corollary obviously suggests an algorithm solving the con-jugacy problem. However, this solution is exponential in general. We in-clude the corollary because of interest in its own. The proof is similar tothat of 6.4.4.

6.4.7 Corollary. Let W be a (finite rank) Coxeter group. Then there

66 Daan Krammer

exists a constant N such that if v, w ∈ W are conjugate, then there existsg ∈W with gvg−1 = w and ℓ(g) ≤ N(ℓ(v) + ℓ(w)).

Proof. Let v, w ∈W be conjugate. Since dΦ(C,Q(w)) = O(ℓ(w)) by 5.6.7,we may assume, after conjugating w by a linearly bounded element, thatC ∩ Q(w) 6= ∅. The proof of 5.10.9 shows that after conjugating w bya bounded element, there exists a general point in Q∗(w) ∩ C. By 5.8.9,Pc(w) is a standard parabolic subgroup. Suppose similarly that Pc(v)is a standard parabolic subgroup. Since v, w are conjugate, so are theirparabolic closures. After conjugating w by a bounded element, we havePc(v) = Pc(w) = WI . Now by 3.1.11, [N(WI) : Z(WI)WI ] < ∞. Hence,after once more conjugating w by a bounded element, we may assume thatv and w are conjugate in WI . We have reduced the theorem to the casewhere Pc(v) = Pc(w) = W and W irreducible.

If W is finite, the result is trivial. If W is affine, the result is a specialcase of 6.4.6. Thus, let us suppose W is infinite and non-affine. As in thebeginning of this proof, conjugate v and w by linearly bounded elementssuch that there are elements x ∈ C ∩ Q(w), z ∈ C ∩ Q(v). Let g ∈ W besuch that gvg−1 = w. Let α be any odd root separating x from wx. Aftermultiplying g at the left by a suitable power of w, the root α also separatesy := gz from wy. By 6.4.2, it follows that for this choice of g, we haveℓ(g) = dΦ(z, gz) + O(1) = dΦ(x, y) + O(1) = O(ℓ(w)). �

6.5 The greatest eigenvalue

In this subsection, we suppose W to be irreducible, infinite, non-affine, Enon-degenerate, Span(Π) = E, and we fix w ∈W such that Pc(w) = W .

We will show in 6.5.14 that there exists λ > 1 such that λ, λ−1 aresimple eigenvalues of w, and such that λ−1 ≤ |µ| ≤ λ for each (complex)eigenvalue µ of w. Moreover, U0 meets the sum of the two correspond-ing (one-dimensional) eigenspaces. The intersection, denoted by Q0(w), isanalogous to the axis of a hyperbolic element of O+(n, 1).

By the theorem of Perron-Frobenius, if an element of GL(E) maps aproper cone into its interior, then it has a simple real eigenvalue boundingevery complex eigenvalue in absolute value. I have been looking for such aproper cone mapped by w into itself, but did not find one. This does notseem to be the right way for showing the desired results. A more directapproach has proved more succesful, and it will be given in this subsection.The most important ingredients are 6.3.8 ((α,wnβ) → ∞ as |n| → ∞whenever α, β are outward roots), 6.3.11 (the number of periodic roots isbounded), and the combinatorially flavoured result 6.8.1 of subsection 6.8.A ‘disadvantage’ of our technique is that we still do not know whether thereexists an eigenvalue µ 6= λ with |µ| = λ. Conjecturally (6.5.16), such a µdoes not exist.

We use the following notation. We will write λ for the maximum of theabsolute values of the complex eigenvalues of w. We denote the generalizedeigenspace of w in E∗ with eigenvalue µ by Eµ; it is defined by Eµ = {x ∈E∗ | (w − µ)kx = 0} where k is big enough, for example, k = dimE∗. Wewrite Ew = Eλ + Eλ−1 . The projections on Eµ and Ew are denoted by pµ

and pw, respectively.

We summarize this subsection. In 6.5.4, it is proved that λ > 1, and


that λ is an eigenvalue of w, and that (w − µ)Eµ = 0 whenever |µ| = λ.Theorem 6.5.10 says pw(U −{0}) = U0 ∩Ew. In 6.5.14, we show dim Eλ =dim Eλ−1 = 1.

6.5.1 Definition. For A ∈ GLn(C), define ρ (A) to be the maximum of theabsolute values of the eigenvalues of A. For µ ∈ C, define deg(µ,A) to bethe maximum among the sizes of the Jordan blocks of A with eigenvalueµ (and 0 if µ is not an eigenvalue). Equivalently, deg(µ,A) is the leastk ≥ 0 such that (A − µ)kx = 0 whenever (A − µ)k+1x = 0. We definedeg(A) = max{deg(µ) | µ ∈ C, |µ| = ρ (A)}.

We will make use of similar notations for sequences a0, a1, . . . of complexnumbers of the form

an =∑

µ∈C∗

cµ(n)µn, (13)

where cµ ∈ C[X] are polynomials, and almost all cµ are identically zero, sothat the above sum is a finite sum. Such sequences are always consideredin the range n ∈ N rather than Z unless stated otherwise.

We call µ an eigenvalue of the sequence if cµ 6= 0. Define ρ (n 7→ an) orsimply ρ (an) to be the maximum of the absolute values of the eigenvaluesof an. We define deg(µ, n 7→ an) or simply deg(µ) to be the degree of cµplus 1 (the degree of the zero polynomial being −1). Finally, deg(an) =max{deg(µ, n 7→ an) | µ ∈ C, |µ| = ρ (an)}.

6.5.2 Lemma. Let (an)n be a non-zero sequence of the form (13). Thenthere is a unique pair (µ, d) ∈ R>0 × Z>0 such that

lim supn→∞

|an|nd−1µn

∈ R>0.

It is µ = ρ (an) and d = deg(an).

Proof. Unicity is easy and left to the reader.Existence. Let µ = ρ (an), d = deg(an). It is easy and left to the reader

to show that the limsup is not infinite. We must show that the limsup isnon-zero.

Let µ0 ∈ C∗ be such that |µ0| = µ and deg(µ0) = d. Choose p0, . . . , pk ∈C such that the sequence bn defined by

bn = p0an + p1an+1 + · · · + pkan+k

has no eigenvalues ν with |ν| = µ, ν 6= µ0, and such that deg(µ0, bn) = d.It is now easy to see that

lim supn→∞

|bn|nd−1µn

6= 0.

It follows that the limsup with an is non-zero too. �

6.5.3 Lemma. Let (an)n be a sequence of the form (13). Let µ = ρ (an)and d = deg(an). If an > 0 for all n, then d = deg(µ).

Proof. Recall that P [X] denotes the set of nonzero polynomials in R[X]whose coefficients are nonnegative. By 5.6.8, there exists f ∈ P [X] such

68 Daan Krammer

that for every eigenvalue λ 6∈ R>0, the polynomial (X − λ)deg(cλ)+1 dividesf . Write f = p0 + p1X + · · · + pkX

k. Then the sequence (bn)n defined by

bn = p0an + p1an+1 + · · · + pkan+k

has only eigenvalues in R>0. Since all an, pi ≥ 0, we have, by the existencepart of 6.5.2

lim supn→∞

|bn|nd−1µn

∈ R>0.

Since all eigenvalues of bn are in R>0, it follows by the unicity part of 6.5.2that d = deg(µ, bn), whence d = deg(µ, an). �

6.5.4 Lemma.(a) λ > 1.(b) Let α, β be two outward roots. Then λ is an eigenvalue of the se-

quence (α,wnβ). In particular, λ is an eigenvalue of w.(c) deg(w) = 1.

Proof. (a). Choose any outward root α. By 6.3.7, there exists m > 0 suchthat (α,wmα) > 1, say (α,wmα) = (µ+ µ−1)/2, µ > 1. By applying 6.2.2inductively to the sequences α < wnmα < w(n+1)mα, we find (α,wnmα) ≥(µn + µ−n)/2. This shows ρ(w) ≥ µ1/m > 1.

(b). For outward roots α, β, write

λαβ = ρ (n 7→ (α,wnβ)), dαβ = deg(n 7→ (α,wnβ)).

First, we will show that λαβ and dαβ do not depend on α, β. Let α, β, α′, β′

be outward. Then there are k, ℓ ∈ Z such that α < wkα′ and wℓβ′ < β by6.3.9. For big n, we have

α < wkα′ < wℓ+nβ′ < wnβ.

By applying 6.2.3 twice, we have

(α,wnβ) ≥ (wkα′, wℓ+nβ′) = (α′, wℓ+n−kβ′). (14)

By 6.3.8 we may write for big n

(α,wnβ) = (an + a−1n )/2, (α′, wnβ′) = (bn + b−1

n )/2, an, bn > 1.

By (14) we havean ≥ bℓ+n−k. (15)

By 6.5.2, it follows that λαβ ≥ λα′β′ , and, by symmetry, all λαβ are equal.By combining (15) and 6.5.2 again, it follows that all dαβ are equal.

Since E is non-degenerate, and the outward roots span E, there existα, β with λαβ = λ. Since (α,wnβ) > 0 for n big, 6.5.3 implies that λ is aneigenvalue of the sequence (α,wnβ).

(c). Let α be an outward root. By 6.3.8, almost all n ∈ N satisfy(α,wnα) > 1. It follows that almost all n ∈ N satisfy (α,wnkα) > 1 for allk > 0; let L be the set of such n.

For n ∈ L, define cn by (α,wnα) = (cn + c−1n )/2, cn > 1. As in the

proof of (a), one finds (α,wnkα) ≥ (ckn + c−kn )/2 for all k ≥ 1. On writing

d = deg(w), we have

lim supk→∞

ckn2(nk)d−1λnk

≤ lim supk→∞

|(α,wnkα)|(nk)d−1λnk

≤ lim supn→∞

|(α,wnα)|nd−1λn

<∞.


Hence cn ≤ λn for all n ∈ L. It follows that

lim supn→∞

|(α,wnα)|nλn

≤ lim supn→∞

λn + λ−n

2nλn= 0,

whence, by 6.5.3, dαα = 1. Since all dαβ are equal, it follows that deg(w) =1, which finishes the proof of (c). �

In words, 6.5.4(c) says that the generalized eigenspace Eµ in E∗ of wwith eigenvalue µ is in fact the eigenspace, if |µ| = λ. The rest of thissubsection is devoted to proving that dim Eλ = 1.

6.5.5 Lemma. LetA ∈ O(k, ℓ), and letEµ denote the generalized eigenspaceof A with eigenvalue µ. Then the following hold.

(a) If µν 6= 1 then Eµ ⊥ Eν .(b) The bilinear form (·, ·) defines a pairing Eµ × Eµ−1 → R if µ2 6= 1.(c) For every µ ∈ C∗, the vector spaces Eµ and Eµ−1 have equal dimen-

sions.

Proof. (a). Let x ∈ Eµ, y ∈ Eν be a counterexample, such that k + ℓ isminimal, where (A− µ)kx = (A− ν)ℓy = 0. Then, using minimality,

(x, y) = (Ax,Ay) = ((Ax− µx) + µx,Ay)

= µ(x,Ay) = µ(x, (Ay − νy) + νy) = µν(x, y),

whence (x, y) = 0. This is the desired contradiction. Part (b) follows from(a), and the fact that (·, ·) is non-degenerate. Part (c) is an immediateconsequence of (b). �

By 6.5.5(c), Eµ and Eµ−1 have equal dimensions, for all µ. Hence, λ−1

is an eigenvalue of w, and every eigenvalue µ satisfies λ−1 ≤ |µ| ≤ λ.

6.5.6 Lemma. Let x ∈ U − {0}, and let α be an outward root. Then λ isan eigenvalue of the sequence 〈x,wnα〉.

Proof. Since x 6= 0, there exists an outward root β such that 〈x, β〉 6= 0.For k > 0 big enough we have 〈x,wkβ〉 ≥ 0. We cannot have equality forall big k > 0, because this would contradict 〈x, β〉 6= 0. Hence there existsk such that 〈x,wkβ〉 > 0. For big n, we have wkβ < wnα by 6.3.9, say(wkβ,wnα) = (an + a−1

n )/2, an > 1. By 1.4.7(c), we find

〈x,wnα〉 ≥ an〈x,wkβ〉.

From 6.5.4(b), it follows that ρ (n 7→ 〈x,wnα〉) = ρ(n 7→ an) = λ. Since〈x,wnα〉 > 0 for n big enough, 6.5.3 implies that λ is an eigenvalue of thesequence 〈x,wnα〉. �

6.5.7 Lemma. For all x ∈ U − {0}, the points pλ(x) and pλ−1(x) arenon-zero.

Proof. This follows immediately from 6.5.6, applied to w and w−1. �

6.5.8 Lemma. On writing r = #S, we have

(1 + w + · · · + wr−1) (U − {0}) ⊂ U0.

70 Daan Krammer

Proof. Let x ∈ U−{0}. Let H = (wZx)′. Then H is a parabolic subgroup,normalized by w. Since Pc(w) = W , we find Pc(N(H)) = W . By 6.8.1, ei-therH is finite orH = W . In the latter case, x = 0 because E is spanned byΠ, a contradiction. Hence H is finite. Note that H = {x,wx, . . . , wr−1x}′.By 2.1.2, the cone X spanned by {x,wx, . . . , wr−1x} contains a point ysuch that y′ = H. By 2.2.5, y ∈ U0. By multiplying y by a scalar andadding another point of X, we find (1 + w + · · · + wr−1)x ∈ U0. �

We recall the notation E>0 for the sum of the generalized eigenspacesin E∗ of w with positive real eigenvalues. Since deg(w) = 1 by 6.5.4(c), wehave pλ(x) = limn→∞wnx/λn for all x ∈ E>0. Applying this result to w−1

as well, we find

pw(x) = limn→∞

(wnx+ w−nx)/λn for all x ∈ E>0. (16)

6.5.9 Lemma. We have pw(Q1(w)) ⊂ U − {0}.

Proof. Let x ∈ Q1(w). Write xn = wnx+ w−nx. Since x ∈ Q1(w) ⊂ E>0,we have (16) pw(x) = limn→∞ xn/λ

n. The proof will be finished if we showthat dΦ(x, xn) is bounded as n→ ∞, since then {x1, x2, . . .} is contained inthe union X of finitely many facets, whence pw(x) ∈ X ⊂ U , and pw(x) 6= 0by 6.5.7.

Since Q1(w) is convex by 5.6.2, we have xn ∈ Q1(w) for all n. A rootα separating x from xn is never even, because if α were even and, say,supporting, then x, xn ∈ Q1(w) ⊂ Q(w) ⊂ H(α). There are only finitelymany periodic roots, by 6.3.11, so we need only consider odd roots. (Infact, periodic roots do not separate x from xn, but we do not need this.)Let α be outward. Since x ∈ E>0 and deg(w) = 1 (6.5.4(c)), there exista, b,M, µ, εk ∈ R such that 1 < µ < λ, and for all k ∈ Z,

〈x,wkα〉 = a λk − b λ−k + εkµ|k|, |εk| ≤M. (17)

We will show that if wkα separates x from xn, then

| a λk − b λ−k| ≤ 2M µ|k|. (18)

We have x ∈ Q(w) ⊂ U0 so x 6= 0. By 6.5.7, a, b 6= 0. Since α is outward,we have a, b > 0. We have

〈xn, wkα〉 = 〈wnx+ w−nx,wkα〉 = 〈x,wk+nα+ wk−nα〉

= (a λk+n − b λ−k−n + εk+nµ|k+n|) + (a λk−n − b λ−k+n + εk−nµ

|k−n|)

= (λn + λ−n) (a λk − b λ−k) + 2 ε µ|n|+|k|, |ε| ≤M. (19)

If wkα separates x from xn, then 〈x,wkα〉 or 〈xn, wkα〉 has different sign

than a λk − b λ−k. In the first case, we find from (17)

| a λk − b λ−k| ≤M µ|k|,

whereas the latter yields by (19)

| a λk − b λ−k| ≤ 2M µ|n|+|k|

λn + λ−n≤ 2M µ|k|,


and either case implies (18). Since (18) has only finitely many solutions ink, the number of roots of the form wkα separating x from xn is boundedas n → ∞. Since the number of wZ-orbits of odd roots is finite, dΦ(x, xn)is bounded as n→ ∞, as promised. �

6.5.10 Theorem. We have pw(U − {0}) = U0 ∩ Ew ⊂ Q1(w).

Proof. Let f ∈ P [X] be a polynomial such that all eigenvalues of f(w) arein R≥0 (see 5.6.9), and such that 1 + X + · · · + XNr−1 divides f , whereN is the constant of 5.6.5. Let g = f(X)f(X−1), h = g(X)/g(λ). Sinceh(λ) = h(λ−1) = 1 and deg(w) = 1, we have pw = pw ◦ h(w) = h(w) ◦ pw.We will show

h(w) (U − {0}) ⊂ Q1(w). (20)

Let x ∈ U −{0}. Write h = pq where q = 1+X + · · ·+Xr−1. By 6.5.8, wehave q(w)x ∈ U0. Moreover, h(w)x ∈ Q2(w) because 1 +X + · · · +XN−1

divides p (see 5.6.5). We even have h(w)x ∈ Q1(w), since all eigenvalues ofh(w) are in R≥0. This concludes the proof of (20).

Now we will prove the theorem. The inclusion pw(U − {0}) ⊃ U0 ∩Ew

is trivial. As to the reverse inclusion, we have

pw(U − {0}) = h(w) ◦ pw ◦ h(w) (U − {0})(20)⊂ h(w) ◦ pw Q1(w)

6.5.9⊂ h(w) (U − {0})(20)⊂ Q1(w) ⊂ U0,

and clearly pw(U) ⊂ Ew. We have shown pw(U − {0}) = U0 ∩ Ew ⊂Q1(w). �

6.5.11 Definition. We define Q0(w) = pw(U − {0}). Thus, by 6.5.10, wehave Q0(w) = U0 ∩ Ew = pw(U0) and Q0(w) ⊂ Q1(w).

Let C(w) denote the conjugacy class of w in W .

6.5.12 Lemma. Let w ∈W be an element in a Coxeter group. Then thereare only finitely many Z(w)-orbits of facets in Q(w).

Proof. It is enough to prove that there are only finitely many Z(w)-orbits ofclosed chambres gC meeting Q(w). We have a bijection Z(w)\W → C(w),Z(w)g 7→ g−1wg. Thus, it remains to prove that ℓ(g−1wg) is bounded aslong as gC meets Q(w). Let x ∈ gC ∩Q(w). By 5.5.7, dΦ(x,wx) = r(w).Hence ℓ(g−1wg) = dΦ(gC,wgC) ≤ dΦ(gC, x)+dΦ(x,wx)+dΦ(wx,wgC) =r(w)+ a bounded number. Hence ℓ(g−1wg) is bounded. �

6.5.13 Lemma. Let α be an outward root. Then Q(w)∩K(−α)∩K(wα)is the union of finitely many facets.

Proof. By 6.5.12, there are only finitely many Z(w)-orbits of facets inQ(w). Since [Z(w) : wZ] < ∞, there are only finitely many wZ-orbits offacets in Q(w). Each orbit has only a finite number of facets contained inK(−α) ∩K(wα), by definition of outwardness, whence the result. �

6.5.14 Theorem. We have dim Eλ = dim Eλ−1 = 1. There exist non-zerou1 ∈ Eλ−1 , u2 ∈ Eλ such that Q0(w) = R>0u1 + R>0u2.

72 Daan Krammer

Proof. Consider the set X = pλ(U0). Note 0 6∈ X by 6.5.7. We will showthat X is open and closed in Eλ − {0}.

Open. This follows from the facts that pλ: E∗ → Eλ is a surjectivelinear map, and that U0 is open in E∗.

Closed. Since pλ = pλpw, we have

X = pλ(U0) = pλ pw(U0) = pλ(Q0(w)).

Let α be an outward root. Then every w-orbit wZx, with x ∈ U0, meetsY := K(−α) ∩K(wα). Hence

X = pλ(Q0(w)) = pλ(Z),

where Z = Q0(w)∩Y . By 6.5.13, and the fact that Q0(w) ⊂ Q(w), the setZ is contained in only finitely many facets. Hence Z ⊂ U , and

X = pλ(Z) ⊂ pλ(U) = pλ(U − {0}) ∪ {0}= pλpw(U − {0}) ∪ {0} ⊂ pλ(U0) ∪ {0} = X ∪ {0},

where in the last ⊂, we use 6.5.10. Hence X is closed in Eλ − {0}.Suppose dim Eλ > 1. Then Eλ−{0} is connected. Since X ⊂ Eλ−{0}

is non-empty, open and closed, we have X = Eλ−{0}. But this contradictsthe fact that X is closed under addition. Hence dim Eλ = 1. Since X isa non-empty cone, there exists u2 ∈ Eλ − {0} such that X = R>0u2.Similarly, pλ−1(U0) = R>0u1 for some u1. We have Q0(w) = pw(U0) =(pλ + pλ−1)U0 ⊂ pλ(U0) + pλ−1(U0) = R>0u1 + R>0u2. So Q0(w) is a non-empty w-invariant convex subset of R>0u1 + R>0u2, that is, the whole ofR>0u1 + R>0u2. �

6.5.15 Proposition. Let α be a root. Then Q0(w) ⊂ µ(α) if and only ifα is periodic.

Proof. ‘If’. Let α be periodic, and let x ∈ Q0(w). Then the sequence〈wnx, α〉 has only eigenvalues λ and λ−1. But this sequence equals 〈x,w−nα〉,and hence has only roots of unity as eigenvalues, since α is periodic. Sinceλ > 1, we find 〈x, α〉 = 0.

‘Only if’. Choose any x ∈ Q0(w). Then all µ(wnα) contain x. Sincethere are only finitely many walls through x, the root α is periodic. �

By definition of λ, for every eigenvalue µ of w, we have |µ| ≤ λ. Thefollowing conjecture sharpens this.

6.5.16 Conjecture. For every eigenvalue µ of w different from λ, we have|µ| < λ. �

6.6 Removed

This subsection has been removed from the thesis, but its number is re-tained to ease reference.

6.7 Second solution to the conjugacy problem

In this subsection, unless stated otherwise, W denotes an infinite, irre-ducible, non-affine Coxeter group, and (E, (·, ·),Π) an associated non-degenerateroot basis with Span Π = E.


We suppose that the root basis is defined over a real number field K(see 4.7.5).

We retain the notations of subsection 6.5. Thus, if some w ∈ W withPc(w) = W has been chosen, we write λ = ρ(w) and Q0(w) = R>0u1 +R>0u2, where wu1 = λ−1u1, wu2 = λu2.

By 6.5.15, Q0(w) is contained in a wall µ(α) if and only if α is peri-odic. Call a point x ∈ Q0(w) special if it is on at least one non-periodic(necessarily odd) wall. Let xk = aku1 + bku2 (k ∈ Z) denote the specialpoints on Q0(w) up to multiples, ordered in the sense that bk/ak is strictlyincreasing. There exists m = m(w) ≤ ℓ(w) such that, after replacing xk bymultiples if necessary, wxk = xk+m.

For every g ∈W , the cone Q0(w)∩gC is the convex span of two, one orno consecutive xk, that is, it is R>0xk +R>0xk+1 or R>0xk or ∅ for some k.

6.7.1 Definition. We define M0(w) to be the set of conjugates g−1wg to wsuch that Q0(w) ∩ gC has dimension 2.

6.7.2 Proposition. There exists a quadratic algorithm that, when givenw ∈W with Pc(w) = W , decides whether w ∈M0(w).

Proof. Let u1, u2, λ be as above. It has to be decided whether Q0(w) ∩ Chas dimension 2, that is, whether there exist at least two t ∈ R>0 such that〈u1 + tu2, es〉 ≥ 0 for all s ∈ S. This is equivalent to a fixed number ofinequalities gi(λ) > 0, where g1, . . . , gn ∈ K[X] have heights O(ℓ(w)) andbounded degrees. By 5.9.2, this can be tested in quadratic time (take f tobe the characteristic polynomial of w). �

6.7.3 Proposition. There exists a cubic algorithm that, given w ∈ Wwith Pc(w) = W , computes an element of M0(w).

Proof. The algorithm runs as follows. In cubic time, compute a pointx ∈ Q(w) of depth O(ℓ(w)), by 5.10.6. Conjugate w so as to get x ∈ C.By 5.10.5, we can compute the canonical outward roots in cubic time. Letα be a canonical outward root. Note that Q0(w) ∩ µ(α) = R>0y for somey ∈ U0. Let y1, . . . , yk be the coordinates of y with respect to a basis of E∗

that was fixed in advance. We may suppose yi = 1 for some i. Let f denotethe characteristic polynomial of w, and λ its greatest real root. Now everycoordinate yi equals a polynomial (depending on yi) over K in λ, of degree≤ k and height O(ℓ(w)). Moreover, dΦ(y, C) = O(ℓ(w)) by 6.4.2. It followsthat the algorithm 1.2.4, applied to y, is cubic (each step is quadratic by5.9.2, and the number of steps is linear — compare 5.10.4). This algorithmcomputes g ∈ W such that y ∈ gC. By conjugating w, we may supposey ∈ Q0(w)∩C. Determine I ⊂ S such that y ∈ CI . Then for some h ∈WI ,we have h−1wh ∈ M0(w). But WI is finite. By simply testing all h ∈ WI

(in quadratic time by 6.7.2), we find an element of M0(w). �

6.7.4 Theorem. For every (finite rank) Coxeter group, there is a cubicsolution to the conjugacy problem.

Proof. In a way similar to 6.4.4, the problem is reduced to the case Pc(v) =Pc(w) = W is irreducible, infinite, non-affine. The algorithm continues asfollows. By 6.7.3, compute g ∈ W such that g−1wg ∈ M0(w), that is,

74 Daan Krammer

gC ∩Q0(w) = R>0x0 +R>0x1, after renumbering the xi if necessary. Thus,we have computed two consecutive special points x0, x1 in Q0(w), that is,we have expressed their coordinates as polynomials (of height O(ℓ(w)) andbounded degree) over K in λ. We compute (in the same sense) x2, x3, . . . asfollows. Suppose xn has been computed. Compute all g ∈W such that xn ∈gC. At least one of these g has the property gC = R>0xn + R>0y for somey independent of xn and independent of xn−1. Then xn+1 may be chosento be y. Computing each next xn takes quadratic time. After computingeach new xn, check whether xn ∈ R>0x0. If so, we have n = m(w), and westop computing any more xi. Now it is easy to compute all of M0(w), inO(m(w)ℓ(w)2) = cubic time. The algorithm is finished by computing oneelement x of M0(v), and checking whether it is in M0(w). This can also bedone in cubic time, because x must be compared to #M0(w) = O(ℓ(w))elements, and each comparison takes quadratic time. �

6.7.5 Remark. Let W ⊂ O+(n, 1) be a hyperbolic reflection group. Theconjugacy problem for hyperbolic elements in W can be solved by the ideasof this subsection. The only point of difference may be 6.7.3.

To explain this, let us denote by L(w) the analogue to Q0(w), that is,L(w) = R>0u1 +R>0u2, where u1, u2 are eigenvectors of w representing thefixed points at infinity. The analogue of 6.7.3 should compute g ∈W suchthat gD ∩ L(w) 6= ∅. Such a g is found by choosing some y ∈ L(w) andthen computing g ∈ W such that y ∈ gD. The algorithm is polynomial ifthe depth dp(y) (see page 54) is polynomial in ℓ(w).

One choice of y could be the projection on L(w) of a point x ∈ D, fixedin advance. We conjecture that dp(y) is O(ℓ(w)) then. I have not provedanything in this direction. Another choice is that Ry is the intersection ofL(w) with an odd wall (see section 5).

Using results of sections 5 and 6, it can then be proved that dp(y) isO(ℓ(w)), analogous to 6.4.2.

6.8 Free abelian groups in Coxeter groups

The algebraic rank of a group G is defined to be the supremum of theranks of the free abelian subgroups of G, and is denoted rk(G). For Coxetersystems (W,S), this should not be confused with the rank #S. In thissubsection, we will compute the algebraic rank of a Coxeter group, in termsof the Coxeter matrix.

6.8.1 Lemma. Suppose W is irreducible. Let I ⊂ S be such that thestandard parabolic closure of N(WI) equals W . Then I is spherical orI = S.

Proof. Suppose I to be non-spherical. Let J be a non-spherical componentof I. By 3.1.9, we have N(WI) = GI ⋉WI . Let g ∈ GI . Then g−1ΠJ = ΠK

for some K ⊂ I. By 3.1.3 there exists a directed path

J = I0s0−→ I1

s1−→ · · · st−→ It+1 = K

in K such that

g = ν(I0, s0) · · · ν(It, st).


The existence of ν(I0, s0) implies that I0 ∪ {s0} contains a spherical com-ponent. Since I0 = J is infinite and irreducible, we must have s0 ∈ J⊥.Hence I1 = J and ν(I0, s0) = s0. Going on this way, we find g ∈WJ⊥ . Weconclude GI ⊂ WJ⊥ , and N(WI) ⊂ WI∪J⊥ . Since the standard parabolicclosure of N(WI) equals W and is contained in WI∪J⊥ , we find I ∪J⊥ = S.But J is a component of I ∪ J⊥ so J = S. Also J ⊂ I so I = S. �

6.8.2 Theorem. Let W be an irreducible, infinite, non-affine Coxetergroup and let H ⊂ W be an abelian subgroup with Pc(H) = W . Then Hdoes not contain a subgroup isomorphic to Z2.

Proof. Suppose that A ⊂ H is a subgroup isomorphic to Z2. Let h ∈A−{1}. We may suppose Pc(h) to be standard parabolic, say Pc(h) = WI .Then

H ⊂ N(WI), (21)

since for all g ∈ H: gWIg−1 = gPc(h) g−1 = Pc(ghg−1) = Pc(h) = WI .

From (21) and Pc(H) = W , it follows that W = Pc(N(WI)). By 6.8.1,we have I = S (note that I is not spherical because h ∈ WI has infiniteorder). Hence Pc(h) = W . This allows us to apply 6.3.10, which states[Z(h) : hZ] < ∞. But Z2 ∼= A ⊂ Z(h). This contradiction finishes theproof. �

For the purpose of fixing an unclarity in the proof of 6.8.3, the statementof 6.8.2 is slightly different from the one in the original thesis, which statedthe following. Let W be an irreducible, infinite, non-affine Coxeter groupand let H ⊂ W be a free abelian subgroup with Pc(H) = W . Then H isinfinite cyclic.

Let I1, . . . , In ⊂ S be irreducible, non-spherical and pairwise perpendic-ular. For all i, let Hi ⊂ WIi

be a subgroup as follows. If Ii is affine, Hi isthe translation subgroup. Otherwise, Hi is any subgroup of WIi

isomorphicto Z. We call

∏iHi a standard free abelian subgroup.

6.8.3 Theorem. Let W be a Coxeter group (of finite rank). Then eachfree abelian subgroup of W has a finite index subgroup which is conjugateto a subgroup of a standard free abelian subgroup.

Proof. Since standard free abelian subgroups of W have bounded rank, itis enough to show that every free abelian subgroup of finite rank verifiesthe theorem. Let G ⊂W be free abelian of finite rank.

After conjugating G if necessary, we may suppose Pc(G) to be standardparabolic, say Pc(G) = WI . Let I1, . . . , In be the components of I. LetGi := G ∩WIi

for all i. Then G is a subgroup of G′ :=∏

iGi ⊂WI .

For all i we shall define a subgroup Hi ⊂ Gi. If Ii is spherical, putHi = 1. If Ii is affine, choose N > 0 such that wN = 1 for all torsionelements w ∈W , and put Hi = {gN | g ∈ Gi}.

Finally, suppose that Ii is non-spherical and non-affine. Then Gi is aninfinite finitely generated abelian group, not containing a subgroup isomor-phic to Z2 by 6.8.2. So Gi contains a finite index subgroup Hi isomorphicto Z; choose any such Hi.

Now H :=∏

iHi is a standard free abelian subgroup and [G′ : G′∩H] <∞ so certainly [G : G ∩H] <∞. �

76 Daan Krammer

A consequence of 6.8.3 is the equivalence (b)⇔(c) of 4.3.1, where Cox-eter groups not containing subgroups isomorphic to Z2 are classified as partof the classification of word hyperbolic Coxeter groups. Our corollary saysa little more, namely, it also characterizes the subgroups isomorphic to Z2,which is not clear from Moussong’s work. Of course, our theory does nottell which Coxeter groups are word hyperbolic. A question is: Can ourtheory be extended so as to classify word hyperbolic Coxeter groups?

Theorem 6.8.3 was subsequently used in [2], [3], [13], [14], [15], [16],[17].

A Euclidean complexes of non-positive curvature

In this appendix we summarize what we need to know about CAT(0) to beable to read appendix B. Our main reference is [9].

A.1 Metric spaces

An interval is a non-empty set I ⊂ R with u, v ∈ I, u < v ⇒ [u, v] ⊂ I.Examples of intervals are (0, 1), (0, 1], (0,∞).

Let (X, d) be a metric space. A path in X is a continuous map froman interval to X. The length λ(α) of a path α: I → X is

sup{∑

i

d(αti, αti+1)∣∣∣ n ≥ 0, t0, . . . , tn ∈ I, ti < ti+1 for all i

}.

A path is said to be normalized if it is of the form α: [0, 1] → X and ofconstant speed c, that is, d(αx, αy) = c d(x, y) for all x, y.

A geodesic in X is a path α: I → X such that d(αx, αz) = d(αx, αy)+d(αy, αz) whenever x < y < z. A geodesic space is a metric space inwhich any two points are connected by a geodesic. A local geodesic isa path α: I → X such that I is a union of open subsets on which α is ageodesic.

A.2 Metric complexes

We denote Euclidean n-space by En and its metric by |x − y| or d(x, y).Denote the n-sphere {x ∈ En+1 : |x| = 1} by Sn and its metric by d(x, y).

A spherical or Euclidean cell is a compact subset of Sn or En, re-spectively, which is the intersection of finitely many closed half-spaces. Bymetric cells we mean spherical or Euclidean cells, and similarly for metriccomplexes, which are to be defined later. The interior of a metric cell X(with respect to the smallest linear variety containing X) is denoted X0. Aface of a metric cell X is either X itself or X ∩H where H is a hyperplanedisjoint with X0.

We collect some properties of metric cells, the proofs of which are leftto the reader. Some properties are proved in [11, § 7]. Let X be a metriccell. Then faces of X are again metric cells. Intersections of faces of Xare again faces of X. The relation ‘being a face of’ is transitive. Everymetric cell is the disjoint union of the interiors of its faces. A metric cellhas only finitely many vertices (= faces with exactly one element). Theconvex hull of a finite subset of Sn or En is a metric cell. All Euclideancells are of this form, but not so for spherical ones. Spherical cells need


not even be connected, as is shown by the spherical cell consisting of twoantipodal points in Sn.

A.2.1 Definition. A Euclidean complex is a pair (K,C) of a set K anda collection C of metric spaces (X, dX), where X ⊂ K (X is called a cell ),such that

MC(1) The cells cover K.MC(2) Every metric space (X, dX) ∈ C is isometric to a Euclidean cell.MC(3) If Y is a face of a cell X, then Y is again a cell, and dY =

dX |Y ×Y .MC(4) The intersection of any two cells is a face of either cell.

A spherical complex is defined as above, by replacing ‘Euclidean’ inMC(2) by ‘spherical’.

For the rest of this subsection, let (K,C) denote a metric complex (thatis, a spherical or Euclidean complex).

Let S be a subset of a cell of K. By MC(3) and MC(4), there is asmallest cell containing S, called the support of S. It is denoted S, andwe shorten {x} to x. For a point x ∈ K, define the star St(x) to be theunion of all cells containing x, and the open star Ost(S) to be the unionof the interiors of all cells containing S.

A linear path in a metric cell X is a map α from an interval I to Xof the following form.

• If X ∈ En is Euclidean: α(t) = x+ ty, x, y ∈ En.• If X ∈ Sn is spherical: α(t) = (cos ωt)u + (sin ωt)v, u, v ∈ Sn ⊂

En+1, u ⊥ v, ω ∈ R≥0, (that is, along a ‘big circle’).

The numbers |y| and ω are called the speed of α.

A linear path in K is a map α from an interval I to K such that α(I)is contained in a cell X, and α is linear with respect to X. A piecewiselinear path is a map α from an interval I to K with the property thatfor all t ∈ I there exists ε > 0 such that α is linear on I ∩ [t − ε, t] andI ∩ [t, t + ε]. Since we will consider only piecewise linear paths, we callthem simply paths. The speed speed(α, t) is defined to be the speed ofa linear restriction of α to an interval containing t. Thus, speed(α, t) isdefined outside a discrete set of values of t. The length λ(α) was definedin subsection A.1 and it is also

∫t∈I speed(α, t)dt.

Recall that a path α: I → K is called normalized if I = [0, 1] andα has constant speed. Note that a path on a compact interval has theform α: [a0, an] → K, where for some a0 < a1 < · · · < an, the restrictionα | [ai, ai+1] is linear for each i. If, in addition, α is normalized, thenα is uniquely determined by the symbol [α(a0), . . . , α(an)], which maytherefore serve as a notation for α. If α: [a, b] → K is a path, we say it isa path from α(a) to α(b).

Define a pseudometric on K, denoted d, as follows. For x, y ∈ K, d(x, y)is the infimum of lengths of paths from x to y. If there is no such path,then d(x, y) = ∞. If x, y are in one cell X, we have d(x, y) ≤ dX(x, y),but equality does not always hold, that is, there may be a shortcut throughother cells. We say y is between x and z if d(x, z) = d(x, y) + d(y, z).So if K is geodesic, then y is between x and z if and only if there exists ageodesic from x to z passing through y.

78 Daan Krammer

By viewing K as the quotient of the disjoint union of cells moduloidentifications, we find a topology on K. Concretely, U ⊂ K is open if andonly if for all cells X, the intersection U ∩ X is open in X (in the usualsense).

Let Shapes(K) denote the set of isometry classes of cells of K.

A.2.2 Theorem. Suppose that Shapes(K) is finite and K is connected.Then K is a complete geodesic space.

Proof. [9, Theorem 7.19]. �

A.3 Curvature

Let X ⊂ Mn = En or Sn be a metric cell, x ∈ X, and let Tx(Mn) denotethe tangent space at x to Mn. We recall that on Tx(Mn) there exists anatural positive definite quadratic form, which we denote by |y|2, and wehave an exponential map exp : Tx(Mn) → E2 defined by: f : t 7→ exp(ty)is a constant speed geodesic and df/dt(0) = y. The link Lk(x,X) andcomplement Cp(x,X) are defined by

Lk(x,X) =

{y ∈ Tx(Mn)

∣∣∣∣|y| = 1; there exists ε > 0such that exp([0, ε]y) ⊂ X

},

Cp(x,X) = {y ∈ Lk(x,X) : y ⊥ Tx(x) ⊂ Tx(Mn)}.

These are spherical cells. If X is a face of Y and x ∈ X, then Lk(x,X) isa face of Lk(x, Y ). For a metric complex K, and x ∈ K, the link Lk(x,K)(or briefly Lk(x)) is defined as the quotient of the disjoint union of linksLk(x,X), X a cell containing x, modulo identifications due to inclusionsof links induced by inclusions of cells. Similarly for complements. Sincelinks and complements depend up to isometry only on the support x of thepoint x, we write Lk(x) for Lk(x) and Cp(x) for Cp(x). The complementCp(X,K) is also defined for the empty cell X, namely, Cp(∅,K) = K.

For all x ∈ K, we have a natural projection St(x) − {x} → Lk(x).

Let S(λ) denote the circle (viewed as a one-dimensional Euclidean orspherical complex) of length λ. A cycle in a metric complex K is a piece-wise linear map α: S(λ) → K. A minimal cycle is a non-constant cyclewhich is a local geodesic.

Roughly, a CAT(κ) space (for κ ∈ R) is a geodesic space whose tri-angles are no fatter than those in the spaces of constant curvature κ. Forthe full definition, which we shall not need, we refer to [9]. For κ = 0 thismeans that we are comparing with Euclidean space En and for κ = 1 withthe metric spheres Sn.

The main aim of this appendix is to recognise from local data if aEuclidean complex is CAT(0), a global condition.

A.3.1 Definition. A metric complex is said to satisfy the link condition iffor every vertex v ∈ K, the link Lk(v,K) is CAT(1). A spherical complexis said to satisfy the girth condition if every minimal cycle has length atleast 2π.

A.3.2 Theorem.


(a) Let K be a Euclidean complex with finitely many shapes. Then Kis CAT(0) if and only if it is simply connected and satisfies the linkcondition.

(b) Let K be a spherical complex with finitely many shapes. Then Kis CAT(1) if and only if it satisfies the link condition and the girthcondition.

Proof. [9, Theorem 5.4]. �

A.3.3 Example. We give an example of a spherical complex which satisfiesthe girth condition but is not CAT(1). It is K = S(λ) ∗ {point} where0 < λ < 2π. Here, ∗ denotes the spherical join; see [9, Definition 5.13] fora definition. Then K contains no minimal cycles, so it satisfies the girthcondition. However, its link at the pole is a circle of length λ < 2π, so Kis not CAT(1).

We now explain the transitivity of complements. Let K be a metriccomplex and let X ⊂ Y ⊂ K be cells. Then Cp(X,Y ) can be embeddednaturally as a cell in Cp(X,K), and

Cp(Y,K) ∼= Cp(Cp(X,Y ),Cp(X,K)). (22)

Conversely, every cell of Cp(X,K) has the form Cp(X,Y ).

A.4 Convex metrics

A function f : R → R is said to be convex if for all x, y ∈ I and all t ∈ [0, 1]we have

f((1 − t)x+ ty) ≤ (1 − t) f(x) + t f(y).

A geodesic space is said to have convex metric if for any two normal-ized geodesics α, β, the function f : t 7→ d(α(t), β(t)) is convex.

A.4.1 Proposition. Every CAT(0) space has convex metric.

Proof. [9, Proposition 2.2]. �

B The Davis-Moussong complex

In the first three subsections of this appendix, we rewrite Moussong’s the-sis, where a Euclidean complex M of non-positive curvature is constructed,on which a Coxeter groups W acts. In subsection B.4, we give an alter-native construction, which is shorter and can be read independently. Insubsection B.5, the metric of M is compared to the distance in W . In sub-section B.6, we derive an exponential solution to the conjugacy problem.

B.1 Nerves of almost negative matrices

An almost negative matrix is a square symmetric real matrix A = (aij)with aij ≤ 0 if i 6= j. Given an almost negative matrix A = (aij), we definea spherical complex N = N(A), called the nerve of A, as follows. We liketo think of the rows and columns of A as indexed by some set I. Let Ebe a real vector space with basis {ei | i ∈ I}. Define a symmetric bilinear

80 Daan Krammer

form (·, ·) on E by (ei, ej) = aij . For a subset J ⊂ I, let EJ denote thesubspace of E generated by {ei | i ∈ J}. We call a subset J ⊂ I sphericalif the bilinear form restricted to EJ is positive definite. For spherical J ⊂ I,define a spherical cell SJ (which is in fact a simplex, that is, the dimensionis the number of vertices minus one) by

SJ ={x =

∑

i∈J

xiei

∣∣∣ xi ≥ 0, |x| =√

(x, x) = 1},

which inherits the structure of a spherical cell by its embedding in theEuclidean space EJ . Now, N(A) is defined to be the spherical complexwith the union of the SJ as underlying point set and the SJ as cells (notingS∅ = ∅).

We call an almost negative matrix normalized if all diagonal elementsare equal to 1. To every almost negative matrix A = (aij)i,j∈I we associatea normalized matrix norm(A) = B, as follows. The rows and columns of Bare indexed by J = {i ∈ I | aii > 0}, and bij = aij/

√aiiajj . The nerves of

A and norm(A) are isomorphic.

B.1.1 Theorem. Nerves of almost negative matrices are CAT(1).

Proof. See [25]. �

B.1.2 Lemma. Let N be the nerve of an almost negative matrix, and letx ∈ N . Then Cp(x,N) is isometric to the nerve of some almost negativematrix.

Proof. Retain the denotations of I, A,E, ei, EJ , so that N = N(A). Wemay assume A to be normalized. By transitivity of complements, we maysuppose x to be a vertex of N , say x = ei. Let fj be the orthogonalprojection of ej on e⊥i , the orthogonal set in E to ei. Thus, for pairwisedifferent indices i, j, k we have

fj = ej − (ej , ei)ei = ej − aijei

and

(fj , fk) = (ej − aijei, ek − aikei)

= ajk − 2 aijaik + aijaikaii = ajk − aijaik ≤ 0

since ajk, aij , aik ≤ 0. Hence the matrix B = (fj , fk)j,k 6=i is again an almostnegative matrix. We claim Cp(ei, N) ∼= N(B). Let EJ be positive definite,with i ∈ J , and let A′, B′ be the restrictions of A, B to J , J − {i},respectively. Let S = {x ∈ EJ : |x| = 1}. Consider the natural mapfrom the tangent space Tei

(S) to e⊥i (which can be defined as translationover −ei, viewing Tei

(S) as a linear variety in EJ). This map restrictsto an isometry Cp(ei, N(A′)) → N(B′) ⊂ e⊥i . Using the fact that for allJ ⊂ I − {i}, the subspace Span{fj | j ∈ J} is positive definite if and onlyif EJ∪{i} is, it follows that Cp(ei, N) ∼= N(B). �

B.2 Cells for the Davis-Moussong complex

For the rest of this appendix, we consider a Coxeter system (W,S), andwe retain the notations of section 1. In this subsection, fix positive realnumbers as (s ∈ S).


Let J ⊂ S be spherical. Since EJ = Span{es | s ∈ J} is non-degenerate,there exists a unique basis {fJ

s | s ∈ J} of EJ dual to {es}, defined by(fJ

s , et) = δst, (s, t ∈ J). Writing Ch for the convex hull, we define

xJ =∑

s∈J

asfJs , XJ = Ch(WJ xJ),

XJK = Ch(WJ xK) for J,K ⊂ S both spherical with J ⊂ K.

The sets XK and XJK are Euclidean cells by their embedding in the Eu-clidean space EK . For spherical J ⊂ S, let pJ : E → EJ denote the or-thogonal projection. It is well defined since the quadratic form on EJ isnon-degenerate.

B.2.1 Definition. For spherical J,K ⊂ S such that J ⊂ K, we definegKJ =

∑s∈K−J f

Ks .

B.2.2 Lemma.(a) dim XJ = #J .(b) Let J ⊂ K be spherical subsets of S. Then pJ xK = xJ . Moreover,

pJ |XJK : XJK → XJ is an isometry of cells, equivariant under WJ .(c) The non-empty faces of XK are precisely all wXJK for all J ⊂ K,

w ∈ WK . For all J ⊂ K, x ∈ XK we have (gKJ , x − xK) ≤ 0, with

equality if and only if x ∈ XJK .

Proof. (a). For all s ∈ J we have s xJ = xJ −2 (xJ , es) es. Since (xJ , es) =as > 0 and since the es (s ∈ J) are linearly independent, the convex hull of{xJ} ∪ {s xJ | s ∈ J} has dimension #J . Hence so has XJ .

(b). E is the orthogonal direct sum of EJ and E⊥J , and WJ fixes E⊥

J

pointwise, and stabilizes EJ . It follows that pJ is WJ -equivariant, and thatpJ sets up an isometry from XJK to its image. By definition of orthogonalprojection, for s ∈ J , we have (pJxK , es) = (xK , es) = as. It follows thatpJxK = xJ and pJXJK = XJ .

(c). From B.2.3, choosing x = xK , p = gKJ , W = WK , we find

(gKJ , wxK − xK) ≤ 0,

with equality only for w fixing gKJ , that is, w ∈WJ by 1.2.2(c). The latter

statement of (c) follows. Thus, XJK is a face of XK . Conversely, each faceof XK containing xK is of the form X = {x ∈ XK | (p, x − xK) = 0},where p ∈ EK , and (p, x − xK) ≤ 0 for all x ∈ XK . The latter conditionimplies, by choosing x = s xK , s ∈ K, that we can write p =

∑s∈J csf

Ks ,

cs > 0 for some subset J ⊂ K. By an argument similar to the above, thevertex set of X is WJxK , whence X = XJK . Note that xK is a vertexof XK because the vertex set of XK is a non-empty subset of WK xK andWK-invariant. Hence every non-empty face of XK is of the form wXJK ,w ∈WK , J ⊂ K. �

B.2.3 Lemma. Let W be a finite Coxeter group, and identify E with itsdual E∗ by (·, ·) = 〈·, ·〉. Let p ∈ C, x ∈ C, w ∈ W . Then (p, wx) ≤ (p, x),with equality only if wp = p.

Proof. Induction on ℓ(w). If w = 1 it is clear. If w = sv, s ∈ S, ℓ(v) <ℓ(w), then by 1.2.2(b) (vx, es) > 0. Hence by induction (p, wx) = (p, vx−

82 Daan Krammer

2 (vx, es) es) = (p, vx) − 2 (vx, es) (p, es) ≤ (p, vx) ≤ (p, x). Now supposeequality holds. Then (p, es) = 0 and (p, vx) = (p, x). Hence sp = p and, byinduction, vp = p. We conclude wp = p. �

B.3 The Davis-Moussong complex

In this subsection, we will construct a Euclidean complex, introduced byMoussong [43, 14.1], and we will prove its main properties B.3.2 and B.3.3.We call it the Davis-Moussong complex, and denote it by M(W ) or M . Adifference between Moussong’s approach and ours is that Moussong buildsthe complex from one fundamental domain, whereas we build it from cells,that is, the approaches are dual.

By a pre-cell we mean a subset of W of the form wWJ , w ∈W , J ⊂ Sspherical. For each pre-cell p, fix an element wp ∈ p. Then p = wpWJ(p)

for a unique spherical J(p) ⊂ S.

B.3.1 Lemma. The intersection of any two pre-cells is either again a pre-cell, or the empty set. Moreover, if p, q are pre-cells with non-empty inter-section, then J(p) ∩ J(q) = J(p ∩ q).

Proof. This follows from 1.2.3(c). �

Let M denote the set of pairs (p, x) where p is a pre-cell and x ∈ XJ(p).

Define an equivalence relation on M by

(p, x) ∼ (q, y) ⇔

r := p ∩ q 6= ∅,

w−1r wpx ∈ XJ(r)J(p), w−1

r wqy ∈ XJ(r)J(q),

pJ(r)(w−1r wpx− w−1

r wqy) = 0.

Now the underlying set of M is M modulo this equivalence relation, andthe cells are the empty set and the images in M of {(p, x) | x ∈ XJ(p)}.Since these sets inject into M , they still carry structures of Euclidean cells.The metric on M is denoted by dM .

B.3.2 Proposition. M(W) is a locally finite Euclidean complex. Thegroup W acts discretely on it, with finitely many orbits of cells. The actionof W on the vertex set is simply transitive.

Proof. Straightforward, using B.2.2 and B.3.1. �

Let v denote the vertex ofM which is the equivalence class of ({1}, x∅) ∈M . Then the neighbours (in the 1-skeleton of M) of v are sv, s ∈ S, and thecorresponding edges have length 2 as. We identify W with the vertex setof M by w 7→ wv. Let YJ denote the images in M of {(WJ , x) | x ∈ XJ}.Thus, the cells of M are the sets wYJ , w ∈W , J ⊂ S spherical.

B.3.3 Theorem (Moussong). M(W ) satisfies the link condition.

Proof. Let A be the Gram matrix of (W,S), that is, A = (ast)s,t∈S , ast =(es, et) = − cos(π/mst). Using 1.2.6, it is easy to see that for every vertexx of M , Cp(x,M) ∼= N(A). From B.1.2 and transitivity of complements,it follows that for every point y ∈ M , the complement Cp(y,M) is the


nerve of some almost negative matrix. By B.1.1, Cp(y,M) satisfies thegirth condition. By A.3.2, M satisfies the link condition. �

B.4 The embedding of the Davis-Moussong complex in the

Tits cone

We will give an alternative definition of the Davis-Moussong complex, byembedding it into U0, the interior of the Tits cone.

Fix a point v ∈ C. We define the underlying point-set of the alternativeDavis-Moussong complex by

M ′ =⋃

{Ch(wWJv) | J ⊂ S spherical , w ∈W}.

The set Ch(WJv) (and thereby Ch(wWJv) for all w ∈ W ) is given thestructure of a Euclidean cell by the embedding q−1r: Ch(WJv) → EJ wherer: E∗ → E∗

J is the restriction map and q: EJ → E∗J is the isomorphism

associated to the inner product (·, ·) on EJ , that is, qx = (x, ·). Let C bethe collection of the cells Ch(wWJv) where w ∈W , J ⊂ S spherical.

B.4.1 Proposition. The pair (M ′, C) is a Euclidean complex, and it isisomorphic to the Davis-Moussong complex M . The edge from v to sv haslength 2〈v, es〉.

Proof. We will prove one step, namely that the intersection of any twocells is a face of either cell. The rest of the proof is straightforward and leftto the reader.

Let X = Ch(gWIv), Y = Ch(hWJv) be cells with non-empty intersec-tion. First we will prove that X and Y have a vertex in common, that is,gWI ∩ hWJ 6= ∅.

Suppose gWI ∩hWJ = ∅, and let x ∈ gWI , y ∈ hWJ such that dW (x, y)is minimal. Let α be a root separating x from y, say 〈xv, α〉 > 0, 〈yv, α〉 <0. Since gWI is gated by 1.3.5(c), and dW (gWI , y) = dW (x, y), we have〈x′v, α〉 > 0 for all x′ ∈ gWI . Hence 〈X,α〉 ⊂ R>0. Similarly 〈Y, α〉 ⊂ R<0,which shows X ∩ Y = ∅, a contradiction. Hence gWI ∩ hWJ 6= ∅.

After left multiplication by some element of W , we may suppose 1 ∈gWI ∩ hWJ , so that X = Ch(WIv), Y = Ch(WJv).

Write K = I ∩ J . We will show X ∩ Y = Ch(WKv). Let y ∈ E∗ suchthat v + y ∈ X ∩ Y . We claim y ∈ Span{sv − v | s ∈ K}.

Let us first show how the claim proves the promised result. We have

v + y ∈ (v + Span{sv − v | s ∈ K}) ∩ Ch(WIv) = Ch(WKv),

where the latter equality follows since Ch(WKv) is a face of Ch(WIv), byB.2.2.

We can write

y =∑

s∈I

as(sv − v) =∑

s∈K

bs(sv − v), as, bs ≥ 0.

In order to prove the claim, we may suppose K = ∅, for otherwise, for eachs ∈ K, subtract min(as, bs) from as and bs, and then replace I and J by

84 Daan Krammer

{s ∈ I | as 6= 0} and {s ∈ J | bs 6= 0}, respectively. Let t ∈ J . We have

〈y, et〉 =∑

s∈I

as〈sv − v, et〉 =∑

s∈I

as〈v, set − et〉

=∑

s∈I

as〈v,−2astes〉 =∑

s∈I

−2 asast〈v, es〉 ≥ 0, (23)

since every term in the last sum is non-negative. In the same way it followsthat

〈y, et〉 =∑

s∈J

−2 bsast〈v, es〉.

In matrix form this reads AJb = c where AJ is the matrix (ast)s,t∈J , andb and c are the vectors b = (−2 bs〈v, es〉)s∈J , c = (〈y, et〉)t∈J . By (23), cis totally non-negative. By 1.5.2, so is A−1

J . Hence so is b = A−1J c. Since

〈v, es〉 > 0 for all s ∈ S, it follows that bs = 0 for all s ∈ J . Hence y = 0,which proves the claim. �

B.5 Comparison of dW and dM

Let M be the Davis-Moussong complex of a Coxeter system (W,S). LetΓ denote the 1-skeleton of M . The object of this subsection is to comparethe path metric of Γ, denoted dΓ, to the restriction of dM to Γ. Clearly, wehave dM ≤ dΓ. Milnor proved (in a more general setting) that there existK,L ∈ R such that dΓ ≤ K dM + L. His proof is non-effective, that is, itdoes not give us K and L explicitly. We will give explicit, though quiteweak bounds for K and L in B.5.6.

Let us identify W with the vertex set in M , such that the left actionsof W coincide. We write ℓΓ(x) = dΓ(1, x) (x ∈ Γ) and ℓM (x) = dM (1, x)(x ∈M).

B.5.1 Lemma. Let N ⊂M be a set containing 1 such that M is coveredby {wN | w ∈ W}. Let F = {f ∈ W | N ∩ fN 6= ∅}. Let there be givenδ,R > 0 such that

N ∩ wN = ∅ ⇒ d(N,wN) ≥ δ for all w ∈W , (24)

ℓΓ(f) ≤ R for all f ∈ F. (25)

Then ℓΓ(x) ≤ (ℓM (x)/δ + 1)R for all x ∈W .

Proof. Let x ∈ W , λ = ℓM (x). Let 1 = x0, x1, . . . , xt = x be a sequenceof points in M such that dM (xi, xi+1) < δ, t ≤ λ/δ + 1. Let hi ∈ W suchthat xi ∈ hiN , h0 = 1, ht = x. Write fi = h−1

i−1hi, so that x = f1f2 · · · ft.Since dM (xi, xi+1) < δ, we have hiN ∩ hi+1N 6= ∅, whence fi ∈ F . HenceℓΓ(x) ≤ ∑

ℓΓ(fi) ≤ tR ≤ (λ/δ + 1)R. �

The following proposition was proved by Milnor [42, Lemma 2].

B.5.2 Proposition. There exist K,L ∈ R such that ℓΓ(x) ≤ K ℓM (x)+Lfor all x ∈W .

Proof. We know that a subset of M is compact if and only if it is boundedand closed, and by B.3.2 that W acts discretely on M with compact quo-tient. It follows easily that the conditions of B.5.1 are fulfilled by certain


N, δ,R, whence the proposition. As an example, we prove the existenceof δ satisfying (24). Suppose N to be compact. Let w1, w2, . . . ∈ W ,d(N,wiN) → 0 (i → ∞), N ∩ wiN = ∅. Since N is bounded, ℓM (wi) isbounded. Since W is discrete in M , the wi have a stable infinite subse-quence. Hence for wi in this subsequence, d(N,wiN) = 0, which impliesN ∩ wiN 6= ∅, a contradiction. �

The above proof is not effective. More precisely, the proof of the ex-istence of δ is non-constructive. We proceed to give an effective, thoughquite weak bound for K and L.

B.5.3 Lemma. Let K be a Euclidean complex. Let there be given δ > 0such that for every cell X of K, and all x, y ∈ X, we have

dX(x, y) = dX(x, y) < δ ⇒ x ∈ y.

Then any two disjoint cells in K have distance at least δ.

Proof. Let X,Y be cells with distance smaller than δ. Let α be a nor-malized geodesic from say x ∈ X to y ∈ Y of length λ(α) = d(X,Y ) < δ.Let 0 = t0 < t1 < · · · < tn = 1 be such that α|[ti, ti+1] is linear, and writexi = α(ti). By induction on i we will prove

α ([0, ti]) ⊂ xi. (26)

For i = 0 this is clear. Let i > 0. Let Z be a cell containing α([ti−1, ti]).We have xi−1 = α(ti−1) ∈ α([ti−1, ti]) ⊂ Z, whence xi−1 ⊂ Z. Hence, byinduction, α([0, ti]) ⊂ Z. Since α is a constant speed geodesic, α|[0, ti] islinear. From λ(α) = d(X,Y ) it follows that λ(α, [0, ti]) = dZ(x, xi). Sincealso λ(α) < δ, we may conclude x ∈ xi. It easily follows that α([0, ti]) ⊂ xi,which proves (26). Choosing i = n in (26), we find Imα ⊂ y. In particular,x = α(0) ∈ y. Hence X ∩ Y 6= ∅, which finishes the proof. �

Note that many complexes do not satisfy the condition of B.5.3 (for allδ > 0). For example, suppose that some cell X is a Euclidean triangle withvertices a, b, c ∈ E2, having a sharp angle at a. Choose y ∈ [a, b] close to a,and let x be the projection of y on [a, c]. Then, if y is close enough to a,dX(x, y) = dX(x, y) < δ but x 6∈ y. The next two lemmas show that theDavis-Moussong complex is special in the sense that B.5.3 can be appliedto it.

Lemma B.5.4 is an effective version of what can be done for every Eu-clidean complex with only finitely many isometry classes of cells. Lemma B.5.5shows a special property of the Davis-Moussong complex, which connectslemmas B.5.3 and B.5.4.

B.5.4 Lemma. Let

δ = min{ 2 as

| gKJ |

∣∣∣ J,K ⊂ S spherical, J ⊂ K, s ∈ K − J}.

Then for every cell X ⊂M , and all x, y ∈ X, we have

d(x, y) < δ ⇒ x ∩ y 6= ∅.

86 Daan Krammer

Proof. Identify X with XK . We may suppose xK ∈ x, say x = XJK ,J ⊂ K. Let us write hK

J = gKJ / | gK

J |, so that |hKJ | = 1. Since for s ∈ K−J

(hKJ , xK − sxK) =

(gKJ , 2 ases)

| gKJ | =

2 as

| gKJ | ,

it follows from the definition of δ that

(hKJ , xK − z) ≥ δ (27)

for every vertex z adjacent to xK , not in XJK . By WJ -invariance of hKJ ,

(27) holds for every vertex z adjacent to a vertex of XJK but which is notin XJK itself. Using the fact that (hK

J , xK − z) = 0 for all z ∈ XJK , iteasily follows that (27) holds for every vertex z of XK which is not in XJK .

Now suppose d(x, y) < δ. Then (hKJ , xK − y) < δ, since |hK

J | = 1.Hence (hK

J , xK − z) < δ for some vertex z of y. Hence z ∈ x ∩ y. �

B.5.5 Lemma. Let K ⊂ S be spherical, and let x, y ∈ XK = Z. Then[dZ(x, y) = dZ(x, y) and x ∩ y 6= ∅

]⇒ x ∈ y.

Proof. We may suppose x ∈ x ∩ y. Write x = XJK , y = XLK . Supposex 6∈ y, so that x 6⊂ y, that is, J 6⊂ L. Let s ∈ J − L.

Since d(x, y) = d(x, y), the projection of y on EJ + xK (= smallestlinear variety containing XJK) is in the interior X0

JK of XJK . Equivalently,pJy ∈ X0

J . In particular, by B.2.2(c),

(fJs , pJ(y − xK)) < 0. (28)

For all t ∈ L we have

(fJs , pJ(t xK − xK)) = (fJ

s , pJ(−2 atet)) = −2 at〈fs, pJet〉 ≥ 0,

by 2.2.2. Since y − xK is a linear combination of t xK − xK (t ∈ L) withnon-negative coefficients, we find (fJ

s , pJ(y − xK)) ≥ 0, contradicting (28).This finishes the proof. �

B.5.6 Proposition. Let δ be as defined in B.5.4. Let R be twice themaximum diameter of ΓJ = Γ ∩ YJ for spherical J ⊂ S. Then ℓΓ(x) ≤(ℓM (x)/δ + 1)R for all x ∈W .

Proof. We will verify the conditions of B.5.1. Choose N to be the union ofall YJ for spherical J ⊂ S. Note f ∈ F ⇔ ∃ spherical J,K ⊂ S: YJ∩fYK 6=∅. Thus, F is the union of all WJWK for spherical J,K ⊂ S. Clearly, (25)is satisfied. We will show that (24) holds. By combination of B.5.4 andB.5.5, the Davis-Moussong complex satisfies the condition of B.5.3. Nowsuppose d(N,wN) < δ. Then d(YJ , wYK) < δ for some spherical J,K ⊂ S.By B.5.3, YJ ∩ wYK 6= ∅, whence N ∩ wN 6= ∅, which proves (24). Theproof is finished by applying B.5.1. �

Probably, B.5.6 is very far from the best possible result. We conjecturethat K,L do not have to depend on W , that is, there exist K,L such thatfor every Coxeter group W , on choosing all edges in M to have length 1,we have dΓ ≤ K dM + L.


B.6 The conjugacy problem

In this subsection, we will show how the Davis-Moussong complex impliesan exponential solution to the conjugacy problem. The same argumentapplies to every non-positively curved group. I heard the idea from M.Shapiro. In [20], a doubly exponential solution, that is, a solution of com-plexity

ABℓ(v)+ℓ(w)

is given, also using the Davis-Moussong complex.

B.6.1 Proposition. There exist computable constants P,Q such that forany two conjugate v, w ∈W , there exists a sequence

v = w0, w1, . . . , wn = w,

such that for all i, we have wi+1 = swis for some s ∈ S, and

ℓ(wi) ≤ P [ℓ(v) + ℓ(w)] +Q.

Proof. Choose the Davis-Moussong complex such that all edges have length 1,and let Γ be its 1-skeleton. Let N = M ∩ C, where we embed M ⊂ U0,and let x ∈ N denote the trivial vertex. Let K,L be computable con-stants such that dΓ ≤ K dM + L (see B.5.6). We will show that P := K,Q := L + 2K diam(N) verify the proposition. Here diam(N) denotes thediameter of N , which is easily shown to be finite. Let v, w ∈ W be conju-gate, say v = g−1wg. Let α, β denote the normalized geodesics in M from xto gx and from wx to wgx = gvx, respectively. By convexity of the metricof M (proposition A.4.1), we have

dM (α(t), β(t)) ≤ (1 − t) dM (x,wx) + t dM (gx, gvx)

≤ (1 − t) ℓ(w) + t ℓ(v) ≤ ℓ(v) + ℓ(w).

ConsiderX := {h ∈W | hN meets Imα}.

Note that 1, g ∈ X. We will show that for all h ∈ X, we have

ℓ(h−1wh) ≤ P [ℓ(v) + ℓ(w)] +Q. (29)

Let h ∈ X. Let y ∈ hN ∩ Imα, say y = α(t), so that wy = β(t). Then

ℓ(h−1wh) = dΓ(hx,whx) ≤ K dM (hx,whx) + L

≤ K [dM (y, wy) + 2 diam(N)] + L

≤ K[ℓ(v) + ℓ(w) + 2 diam(N)] + L,

which shows (29). For every point y ∈ M , the set {h ∈ W | y ∈ hN} isconnected. It is easy to deduce that X is connected, which concludes theproof. �

The above proposition suggests the following algorithm solving the con-jugacy problem. Let G be the graph with vertex set

{g ∈ C(w) | ℓ(g) ≤ P [ℓ(v) + ℓ(w)] +Q}where g, h are adjacent if and only if sgs = h for some s ∈ S. Compute theconnected component G0 of G containing w, by walking through G0. Thetime needed for doing so is exponential in ℓ(v) + ℓ(w), since #G0 ≤ #G isexponential. By B.6.1, v and w are conjugate if and only if v ∈ G0.

88 Daan Krammer

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[48] E.B. Vinberg, Hyperbolic reflection groups. Russian Math. Surveys40:1 (1985), 31–75. [4]


Index

affine Coxeter group . . . . . . . . . . . . 29algebraic rank . . . . . . . . . . . . . . . . . . 74axis . . . . . . . . . . . . . . . . . . . . . 33, 39, 71canonical root . . . . . . . . . . . . . . . . . . 54Cayley graph . . . . . . . . . . . . . . . . . . . 12centralizer . . . . . . . . . . . . . . . 26, 43, 61cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76classical root basis . . . . . . . . . . . . . . 13combinatorially convex . . . . . . . . . 50commensurable . . . . . . . . . . . . . . . . . . 5complement . . . . . . . . . . . . . . . . . . . . 78complex . . . . . . . . . . . . . . . . . . . . . . . . 77complexity . . . . . . . . . . . . . . . . . . . . . 4,6component . . . . . . . . . . . . . . . . . . . . . 18cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20conjugacy problem . . . . . . . . . . . . . . . 4convex . . . . . . . . . . . 14, 16, 45, 79, 79convex metric . . . . . . . . . . . . . . . . . . 79Coxeter

graph . . . . . . . . . . . . . . . . . . . . . . 12group . . . . . . . . . . . . . . . . . . . . . . 12matrix . . . . . . . . . . . . . . . . . . . . . 12system . . . . . . . . . . . . . . . . . . . . . 12

critical . . . . . . . . . . . . . . . . . . . . . . . . . 49cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . 54depth . . . . . . . . . . . . . . . . . . . . . . . . . . 54dimension of a cone . . . . . . . . . . . . . 20dimension of a facet . . . . . . . . . . . . 14dual

vector space . . . . . . . . . . . . . . . 13cone . . . . . . . . . . . . . . . . . . . . . . . 20

elliptic elements of O+(n, 1) . . . . 32equivalence of odd roots . . . . . . . . 60essential . . . . . . . . . . . . . . . . . . . . 49, 49Euclidean . . . . . . . . . . . . . . . . . . . . . . 76even . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36face . . . . . . . . . . . . . . . . . . . . . . . . . 13, 76facet . . . . . . . . . . . . . . . . . . . . . . . . . . . 14facial subset . . . . . . . . . . . . . . . . . . . . 13facial subgroup . . . . . . . . . . . . . . . . . 21Galois correspondence . . . . . . . . . . 20gated . . . . . . . . . . . . . . . . . . . . . . . . . . . 16general point in Q(w) . . . . . . . . . . 51general point in Q∗(w) . . . . . . . . . 52geodesic in a complex . . . . . . . . . . . 76girth condition . . . . . . . . . . . . . . . . . 78glide-reflection . . . . . . . . . . . . . . . . . . 37Gram matrix . . . . . . . . . . . . . . . . . . . 13halfspace . . . . . . . . . . . . . . . . . . . 14, 21height . . . . . . . . . . . . . . . . . . . . . . . . . . 53hyperbolic

elements of O+(n, 1) . . . . . . . 32reflection group . . . . . . . . . . . . 30

space . . . . . . . . . . . . . . . . . . . . . . 31infinity, points at . . . . . . . . . . . . . . . 31interval . . . . . . . . . . . . . . . . . . . . . . . . . 76inward . . . . . . . . . . . . . . . . . . . . . . . . . 44irreducible . . . . . . . . . . . . . . . . . . 13, 20join . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79length . . . . . . . . . . . . . . . . . . . . . . . . . . 12length of a path . . . . . . . . . . . . . . . . 76linear path . . . . . . . . . . . . . . . . . . 77, 77link . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78link condition . . . . . . . . . . . . . . . . . . 78local geodesic . . . . . . . . . . . . . . . . . . . 76longest element . . . . . . . . . . . . . . . . . 15metric cell . . . . . . . . . . . . . . . . . . . . . . 76minimal cycle . . . . . . . . . . . . . . . . . . 78nerve . . . . . . . . . . . . . . . . . . . . . . . . . . . 79non-negative matrix . . . . . . . . . . . . 20normalized . . . . . . . . . . . . . . . . . . 76, 80normalizer . . . . . . . . . . . . . . . . . . . . . . 26number field . . . . . . . . . . . . . . . . . . . . 53odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36open star . . . . . . . . . . . . . . . . . . . . . . . 77ordering on roots . . . . . . . . . . . . . . . 19outward . . . . . . . . . . . . . . . . . . . . . . . . 44parabolic

closure . . . . . . . . . . . . . . . . . 22, 51elements of O+(n, 1) . . . . . . . 32root subsystem . . . . . 15, 28, 62subgroup . . . . . . . . . . . . . . . . . . 13

periodic . . . . . . . . . . . . . . . . . . . . . . . . 35pointed cone . . . . . . . . . . . . . . . . . . . . 20proper action . . . . . . . . . . . . . . . . . . . . 5proper cone . . . . . . . . . . . . . . . . . . . . 20pseudometric . . . . . . . . . . . . . . . . . . . 38quadratic . . . . . . . . . . . . . . . . . . . . . . . 54rank

of a Coxeter system . . . . . . . . 12of a group . . . . . . . . . . . . . . . . . 74

reflection . . . . . . . . . . . . . . . . . . . . . . . 15reflection subgroup . . . . . . . . . . . . . 17rest root . . . . . . . . . . . . . . . . . . . . . . . . 49root . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

basis . . . . . . . . . . . . . . . . . . . . . . . 13subbasis . . . . . . . . . . . . . . . . . . . 18subsystem . . . . . . . . . . . . . . . . . 17system . . . . . . . . . . . . . . . . . . . . . 14

separate . . . . . . . . . . . . . . . . . . . . 15, 39signature . . . . . . . . . . . . . . . . . . . . . . . 29solid cone . . . . . . . . . . . . . . . . . . . . . . 20special point in Q0(w) . . . . . . . . . . 73speed . . . . . . . . . . . . . . . . . . . . . . . . . . . 77spherical . . . . . . . . . . . . . . . . . . . . . . . 15spherical cell or complex . . . . . . . . 76

92 Daan Krammer

spherical join . . . . . . . . . . . . . . . . . . . 79stable . . . . . . . . . . . . . . . . . . . . . . . . . . 20stable closure . . . . . . . . . . . . . . . . . . . 20standard parabolic

closure . . . . . . . . . . . . . . . . . . . . . 14root subsystem . . . . . . . . . . . . . 15subgroup . . . . . . . . . . . . . . . . . . 13

star . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77straight . . . . . . . . . . . . . . . . . . . . . . . . 35straight line length . . . . . . . . . . . . . . 6support . . . . . . . . . . . . . . . . . . . . . . . . 77supporting . . . . . . . . . . . . . . . . . . . . . 45symbol . . . . . . . . . . . . . . . . . . . . . . . . . 77Tits cone . . . . . . . . . . . . . . . . . . . . . . . 13transitivity of complements . . . . . 79universal Coxeter group . . . . . . . . 37wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21word hyperbolic group . . . . . . . . . . 30word problem . . . . . . . . . . . . . . . . . . . . 4


List of symbols

Also see page 3 for some common notation.

as . . . . . . . . . . . . . . . . . . . . . . . . . . . 80ast = (es, et) . . . . . . . . . . . . . . . . . 13AI , Gram matrix . . . . . . . . . . . . 13A(α), halfspace . . . . . . . . . . . . . . 14

c . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37C, fundamental chamber . . . . 13CI . . . . . . . . . . . . . . . . . . . . . . . . . . 13C(w), centralizer . . . . . . . . . . . . 71Cp, complement . . . . . . . . . . . . . 78

d, metric on Hn . . . . . . . . . . . . . 31d, metric on a complex . . . . . . 77deg(λ,w), deg(w) . . . . . . . . . . . 67deg(λ, an), deg(an) . . . . . . . . . . 67dW , metric on W . . . . . . . . . . . . 12dα, dΦ . . . . . . . . . . . . . . . . . . . . . . . 39D,DI . . . . . . . . . . . . . . . . . . . . . . . 32DIJ . . . . . . . . . . . . . . . . . . . . . . . . . 16

es . . . . . . . . . . . . . . . . . . . . . . . . . . . 13E . . . . . . . . . . . . . . . . . . . . . . . . . . . 13E∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 13EI . . . . . . . . . . . . . . . . . . . . . . . . . . 13Eµ, Ew . . . . . . . . . . . . . . . . . . . . . . 66E>0 . . . . . . . . . . . . . . . . . . . . . . . . . 45En, Euclidean space . . . . . . . . . . 76

fs . . . . . . . . . . . . . . . . . . . . . . . . . . . 13fJ

s . . . . . . . . . . . . . . . . . . . . . . . . . . 80fα . . . . . . . . . . . . . . . . . . . . . . . . . . . 38F (X), free group . . . . . . . . . . . . . 4

gKJ . . . . . . . . . . . . . . . . . . . . . . . . . . 81GI . . . . . . . . . . . . . . . . . . . . . . . . . . 26

H . . . . . . . . . . . . . . . . . . . . . . . . . . . 31H+, H− . . . . . . . . . . . . . . . . . . . . . 31Hn, hyperbolic space . . . . . . . . 31H(α), H(A) . . . . . . . . . . . . . . . . . 19

I⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . 18I

s−→ J . . . . . . . . . . . . . . . . . . . . . 24

J(p) . . . . . . . . . . . . . . . . . . . . . . . . . 82

KI . . . . . . . . . . . . . . . . . . . . . . . . . . 21K(α),K(A) . . . . . . . . . . . . . . . . . 19

K . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

ℓ . . . . . . . . . . . . . . . . . . . . . . . . 12, 30L(w) . . . . . . . . . . . . . . . . . . . . . . . . 74Lk, link . . . . . . . . . . . . . . . . . . . . . 78

mst, order of st . . . . . . . . . . . . . . 12m(w) . . . . . . . . . . . . . . . . . . . . . . . . 73M = M(W ) . . . . . . . . . . . . . . . . . 82M0(w) . . . . . . . . . . . . . . . . . . . . . . 73

N(WI), normalizer . . . . . . . . . . 26N(A), nerve . . . . . . . . . . . . . . . . . 79

Ost(x), open star . . . . . . . . . . . . 77Out(w) . . . . . . . . . . . . . . . . . . . . . . 44O+(n, 1) . . . . . . . . . . . . . . . . . . . . 31

pµ, pw . . . . . . . . . . . . . . . . . . . . . . . 66pI . . . . . . . . . . . . . . . . . . . . . . . . . . . 22P [X] . . . . . . . . . . . . . . . . . . . . . . . . 47Pc(X), parabolic closure . . . . 22Pc∞(w), Pc0(w) . . . . . . . . . . . . . 52PSn . . . . . . . . . . . . . . . . . . . . . . . . . 31

qI . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Q(w), axis . . . . . . . . . . . . . . . . . . 39Q0(w) . . . . . . . . . . . . . . . . . . . . . . . 71Q1(w), Q2(w) . . . . . . . . . . . . . . . . 45Q∗(w) . . . . . . . . . . . . . . . . . . . . . . . 50Q+(w) . . . . . . . . . . . . . . . . . . . . . . 53Q(x) = (x, x) . . . . . . . . . . . . . . . . 31

r = #S . . . . . . . . . . . . . . . . . . . . . 37rk(G) . . . . . . . . . . . . . . . . . . . . . . . 74r(w) . . . . . . . . . . . . . . . . . . . . . . . . 44rα, reflection . . . . . . . . . . . . . 13, 14rA . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Rg . . . . . . . . . . . . . . . . . . . . . . . . . . 33

S, generators of W . . . . . . . . . . 13S, S+, S−, Sn−1 . . . . . . . . . . . . 31S(λ), circle . . . . . . . . . . . . . . . . . . 78Sn, n-sphere . . . . . . . . . . . . . . . . . 76St(x), star . . . . . . . . . . . . . . . . . . . 77SJ . . . . . . . . . . . . . . . . . . . . . . . . . . 80

U , Tits cone . . . . . . . . . . . . . . . . . 14

94 Daan Krammer

U0, interior of U . . . . . . . . . . . . 14

wI , longest element . . . . . . . . . . 15wp . . . . . . . . . . . . . . . . . . . . . . . . . . 82W . . . . . . . . . . . . . . . . . . . . . . . . . . . 13WI . . . . . . . . . . . . . . . . . . . . . . . . . . 13

x, support . . . . . . . . . . . . . . . . . . . 77xJ , XJ . . . . . . . . . . . . . . . . . . . . . . 81X0, interior of a cell . . . . . . . . . 76XJK . . . . . . . . . . . . . . . . . . . . . . . . 81

YJ . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Z(w), centralizer . . . . . . . . . . . . 43Z(WI), centralizer . . . . . . . . . . . 26

λ = ρ (w) . . . . . . . . . . . . . . . . . . . . 66λ(e) . . . . . . . . . . . . . . . . . . . . . . . . . 25

µ(α), wall . . . . . . . . . . . . . . . . . . . 21µ(A), wall . . . . . . . . . . . . . . . . . . . 39

ν(I, s) . . . . . . . . . . . . . . . . . . . . . . . 24

π . . . . . . . . . . . . . . . . . . . . . . . . 30, 31Π . . . . . . . . . . . . . . . . . . . . . . . . . . . 13ΠI . . . . . . . . . . . . . . . . . . . . . . . . . . 13

ρ (w) . . . . . . . . . . . . . . . . . . . . . . . . 67ρ (an) . . . . . . . . . . . . . . . . . . . . . . . 67

φ . . . . . . . . . . . . . . . . . . . . 26, 30, 33Φ, root system . . . . . . . . . . . . . . 14ΦI . . . . . . . . . . . . . . . . . . . . . . . . . . 15Φodd . . . . . . . . . . . . . . . . . . . . . . . . 52

ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

(·, ·) . . . . . . . . . . . . . . . . . . . . . . . . . 13[x, y] . . . . . . . . . . . . . . . . . . . . . . . . 14<, ordering on roots . . . . . . . . . 19′ prime map . . . . . . . . . . . . . . . . . 20∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Documents

The conjugacy problem for Coxeter groups - Warwick Insitemasbal/index_files/me.pdf · 2008-06-18 · The conjugacy problem for Coxeter groups Daan Krammer June 18, 2008 University