Some Groups Having Only Elementary Actions on Metric Spaces with Hyperbolic Boundaries

Some Groups Having Only Elementary Actions on

Metric Spaces with Hyperbolic Boundaries?

ANDERS KARLSSON1 and GUENNADI A. NOSKOV21FIM, ETH-Zentrum, 8092 Zurich, Switzerland. e-mail: [email protected] of Mathematics, University of Bielefeld, 33501 Bielefeld, Germany;

Institute of Mathematics, Omsk, 644099 Pevtsova 13, Russia.e-mail: [email protected]

(Received: 26 August 2002)

Abstract. We study isometric actions of certain groups on metric spaces with hyperbolic-typebordifications. The class of groups considered includes SLnðZÞ, Artin braid groups and map-

ping class groups of surfaces (except the lower rank ones). We prove that in various ways suchactions must be elementary. Most of our results hold for non-locally compact spaces andextend what is known for actions on proper CAT(-1) and Gromov hyperbolic spaces. We also

show that SLnðZÞ for n5 3 cannot act on a visibility space X without fixing a point in X .Corollaries concern Floyd’s group completion, linear actions on strictly convex cones, andmetrics on the moduli spaces of compact Riemann surfaces. Some remarks on boundedgeneration are also included.

Mathematics Subject Classifications (2000). primary: 20Fxx, 20Hxx, 51K99; secondary:22E40, 58B20, 32G15.

Key words. convergence groups, hyperbolic spaces, mapping class groups, non-locally

compact metric spaces, superrigidity, visibility manifolds.

1. Introduction

The philosophy of this paper is somewhat similar to Steinberg’s article [42], in which

he deduces significant special cases of important theorems for some matrix groups

‘by using certain obvious relations among the simplest elements of these groups in

rather simple-minded ways’. In our setting, we also do not treat general lattices in

some class of topological groups as is common in rigidity theory, on the other hand

we are able to incorporate important groups arising in other contexts. In contrast to

[42] however, our results are of mainly one type: degeneracy of actions on metric

spaces with hyperbolic boundaries.

Although several of the mechanisms behind our results are well-recognized, a few

novel points in our proofs allow for a more general treatment. The metric spaces

considered here are not necessarily locally compact, CAT(0), geodesic or d-hyper-

?Research partly supported by the DFG research group ‘Spektrale Analysis, asymptotische

Verteilungen und stochastische Dynamik’.

Geometriae Dedicata 104: 119–137, 2004. 119# 2004 Kluwer Academic Publishers. Printed in the Netherlands.

bolic, and our arguments may also prepare the ground for a more thorough study of

actions on certain infinite dimensional spaces advocated for example in [19, pp. 120–

122]. In some situations (e.g. SLn53ðZÞ-actions on classical hyperbolic spaces) we

reprove some very special cases of the more sophisticated theory of superrigidity

([31]). At other instances (e.g. Theorem 3), one might say that we contribute to this

theory. Corollaries 1–4 below are some consequences of our considerations. Even

though actions on trees are included in our study, the reader should consult Culler

and Vogtmann [12] for more precision in this case.

We propose a notion of metric space with hyperbolic bordification, which include

CAT(-1)-spaces, visibility manifolds, Gromov’s d-hyperbolic spaces, and Hilbert’smetric spaces of strictly convex domains with their standard boundaries, as well

as, graphs with the end-compactification and general geodesic metric spaces comple-

ted following Floyd and Gromov. See Section 2 for the definition and Section 7 for

more details on the examples.

Let us now proceed to define the classes of groups involved in this paper. Let S be

a subset of group G and P a property of groups. Define a graph, called the P-graph ofS, which has S as the vertex set and two vertices s1 and s2 are joined by an edge if and

only if the group generated by s1 and s2 has the property P. For P, we will consider

the properties of: commutation (C ), nilpotency (N ), not containing a non-Abelian

free semigroup (NoFS2), and finally, not containing a non-Abelian free subgroup

(NoF2).

Let A (respectively, B) be the class of groups which are generated by a set S whoseNoFS2-graph is connected (respectively, whose NoF2-graph is connected). Clearly,

A � B. When we speak of the generators of a group in these classes, we will alwaysmean the elements in the special generating set S.

Examples include many amenable groups, products of infinite groups, SLnðZÞ and

SLnðRÞ for n5 3, braid groups Bn for n5 5, mapping class groups Mod(Sg) for

g5 2, and some automorphism groups of free groups SAutðFnÞ, for n5 5. See

the last section for more details.

A sequence of isometrics fgng1n¼0 is called unbounded if dðgnx; xÞ is unbounded

in n for some (or any) x, and a single isometry g is called unbounded if gn :¼ gn is

unbounded. An action is called metrically proper if whenever gn leaves every finite

subset of G, it holds that dðx; gnxÞ ! 1 for some (or any) x. An action is called

elementary (following, e.g., ([18]) if the limit set of any orbit consists of at most

two points. (Note that some authors define an action to be elementary if the whole

group fixes a point on the boundary or in the space.)

Let X be a hyperbolic bordification, or more generally, a contractive bordification

(see Section 2) of a complete metric space X and such that the isometries of X extend

to homeomorphisms of X . We obtain:

THEOREM 1. Let G be a group in A and assure that G acts on X by isometries such

that the elements in S are unbounded. Then G fixes a point in @X. If there are two fixed

points, then the action is in addition elementary.

120 ANDERS KARLSSON AND GUENNADI A. NOSKOV

THEOREM 2. Let G be a group is B and assume that G acts on X by isometries such

that the elements in S are unbounded. Suppose, moreover, that every two elements in S

generate a metrically proper action. Then the action of G is elementary and G fixes a

point in @X.

For example, X can be a d-hyperbolic space, CAT(-1)-space, or a locally visibleCAT(0)-space with their usual bordifications, see [8] for details. Note that the

hypotheses in Theorem 2 are clearly satisfied if every generator has infinite order

and the action by G is metrically proper. These theorems are best possible in thesense that any hyperbolic Coxeter group (which indeed belongs to A) acts non-trivially on its Cayley graph but each generator certainly acts boundedly.

Combining Theorem 1 with the fact that SLðn;ZÞ; n5 3, is boundedly generated

by elementary matrices (see Sections 6 and 8) we answer the question formulated in

the Introduction of Fujiwara’s paper [15]:

THEOREM 3. Any isometric action of SLðn;ZÞ; n5 3, on a uniformly convex

complete metric space X with a contractive bordification ðe.g. a visibility manifoldÞ

must have a global fixed point inX .

For a statement without the assumption of uniform convexity we refer to Section 6.

As already indicated above, our results have a large overlap with previously

known results; we now try to briefly make some references to these. The theory

of Fuchsian and Kleinian groups is concerned with discrete groups of isometries

of real hyperbolic spaces and in these situations the statements in Sections 3 and

4 are classical. The phenomenon of certain groups not admitting nondegenerate

actions on certain spaces occurs frequently in rigidity theory. One milestone here

is Margulis’ work [31] with its impressive generality and depth. This work has

been extended in various directions by many people, we mention only Burger

and Mozes [10], Gao [16], Fujiwara [15], and Monod and Shalom [35] as these

papers consider the setting of CAT(-1)-spaces (or visibility manifolds in the case

of [15]) and are therefore most directly related to the present discussion. For trees

there is the theory of Bass and Serre [38], and further contributions by Tits [44],

Margulis [32], Alperin [2], and Watatani [45]. Among more recent developments

for trees, we have Bogopolski [6], Pays and Valette [37], Noskov [36], Lubotzky,

Mozes and Zimmer [30], Shalom [39] and Culler and Vogtmann [12]. There

are of course many other related and notable works that could be mentioned.

Except in the tree-case and for [16] nonlocally compact situations are usually

not treated.

Many groups do admit elementary proper actions on some CAT(-1)-space in view

of a simple warped product construction, see [19, p. 157]. On the other hand, it

follows from [28, Corollary 6.2] that an action of a nonamenable group on a

CAT(0)-space with orbit function growing at most exponentially (e.g., a properly dis-

continuous action on a locally compact Cartan-Hadamard manifold with curvature

SOME GROUPS HAVING ONLY ELEMENTARY ACTIONS 121

bounded from below) always has an infinite limit set, hence cannot be elementary.

In view of Theorem 2 we therefore get (keep in mind that the fundamental group

of a nonpositively curved manifold is always torsion-free):

COROLLARY 1. A nonamenable group in the class B cannot be the fundamental

group of a manifold which admits a complete metric for which the universal cover is a

visibility manifold with at most exponential volume growth.

More specifically, we also have:

COROLLARY 2. Let T be the Teichmuller space of a surface of genus g and with n

punctures. Assume that 3g 3þ n5 3: Then there is no invariant complete CAT(0)-

metric for which T is a visibility space and such that the number of orbit points of

Modg;n in balls grows at most exponentially in the radius.

Theorem 2 (at least for X a locally compact d-hyperbolic space) and Corollary1 (at least for pinched negative curvature) for the mapping class groups (g5 2)

were already known to McMullen [34, p. 327] and Brock and Farb [9]. In the

latter reference several other interesting and related theorems about invariant

metrics on Teichmuller spaces are proved. Our Corollary 1 generalizes Theorem

1.3 in [9]. Note the contrast of these results with the fact that the Weil–Petersson

metric on the moduli spaces is a noncomplete(!) metric of negative (but not pin-

ched!) curvature and the result of Masur and Minsky [33] that the complex of

curves, on which Mod(Sg;n), g5 1, acts unbounded (but not properly!), are

d-hyperbolic.Another application of Theorem 2 one can get employing Hilbert’s metric on

convex sets is:

COROLLARY 3. Assume that a group G in class B with infinite order generators acts

properly discontinuously by projective automorphisms on a strictly convex bounded

domain C in RN. Then for any v 2 C, the limit set Gv \ @C consists of at most two points.

As a finitely generated group G acts metrically properly by isometries on itself anda Floyd boundary @G, which is a hyperbolic boundary, equals the limit set LðGÞ, weget from Theorem 2:

COROLLARY 4. Any Floyd boundary of a finitely generated group in the class Bwith infinite order generators consists of at most two points.

This provides more examples for the discussion in the last paragraph of [14] and

has a certain consequence for harmonic functions on such groups, see [27]. We refer

to the article of Gromov and Pansu [20] for a discussion on Floyd boundaries, see

also [19] and [26].


2. Hyperbolic Bordifications

Let (X, d ) be a complete metric space. A bordification of X is a Hausdorff topolo-

gical space X with X embedded as an open dense subset. The boundary is @X :¼X nX.

Fix a point x0 2 X and following Gromov and Lyndon, we define for any two

points z;w 2 X:

ðzjwÞ :¼ 12ðdðz; x0Þ þ dðw; x0Þ dðz;wÞÞ:

For a set W �X , we let (ðxjWÞ :¼ supfðxjwÞ : w 2 W \ Xg.

The spaceX is called a hyperbolic bordification of X if:

HB 1 For any x 2 @X, there is a countable family of neighborhoods W of x inX ,such that the collection of open sets

fx : ðxjWÞ > Rg [W;

whereW 2 W and R > 0, constitutes a fundamental system of neighborhoods of

x in X .

HB 2 Any sequence fxng in X for which ðxnjxmÞ ! 1 as n;m ! 1, converges to a

point in @X.

Note that HB 1 implies HB 2 when X is sequentially compact. (In the case X is a

geodesic space, we could alternatively use the expression dðx0; ½z;w�Þ instead of ðzjwÞ,

although it seems that in general this would not be quite equivalent.)

The one-point bordification is a trivial hyperbolic bordification of any complete

metric space. For nontrivial examples, see Section 7.

An Isom(X )-bordification is a bordification of X, where the action of IsomðX Þ on

X extends continuously to an action by homeomorphisms of X and @X. A contractive

bodification is an Isom (X )-bordification which satisfies:

CB 1 (Contractivity) whenever gn 2 IsomðX; d Þ such that gnx0 ! z 2 @X and

g1n x0 ! Z 2 @X, then gnz ! z for every z 2X nfZg and this convergence isuniform outside every neighborhood of Z.

CB 2 For any unbounded isometry g, there are numbers nk such that gnkx0 and

gnkx0 converge to some point(s) in @X as k ! 1.

CB 3 For any unbounded sequence gn such that gnx ! x for some x 2 @X, there is a

subsequence gnk such that at least one of gnkx0 and g1nk x0 converges to x ask ! 1.

The first conditions is taken from [46] and implies the last two conditions for a

sequential compact X . Let us emphasize that the axioms CB 2 and CB 3, which

may not have been much considered previously, play a crucial role in this paper.

They are justified by the following theorem.

THEOREM 4. Let X be a hyperbolic Isom ðX Þ-bordification of a complete metric

space X. Then X is a contractive bordification.


Proof. The proof of CB 1 follows a nice argument of Woess [46]:

Let gn be a sequence of isometries with gnx0 ! z and g1n x0 ! Z. Let U and V

be neighborhoods in X of z and Z respectively. By HB 1 we can find R > 0, and

neighborhoods W and W 0 of z and Z respectively such that

A :¼ fx : ðxjWÞ > Rg � U

B :¼ fx : ðxjW 0Þ > Rg � V:

For any z outside B we have

ðgnx0jgnzÞ ¼12ðdðx0; gnx0Þ þ dðx0; gnzÞ dðz; x0ÞÞ

¼ dðx0; gnx0Þ 12ðdðx0; gnx0Þ dðg

1n x0; zÞ þ dðz; x0ÞÞ

¼ dðx0; gnx0Þ ðg1n x0jzÞ5 dðx0; gnx0Þ R

for all n such that gnx0 2 W and g1n x0 2 W 0. Therefore we have that there is N > 0

such that

gnz 2 A � U

for all n5N and every z 2 Bc � Vc. Since isometries extend continuously to the

boundary CB 1 is proved.

Assuming HB 2, the proof of CB 2 is essentially given in [24, p. 1454], or see [26,

Prop. 4]. The proof of CB 3 basically follows the proof of Lemma 2 in [26] (this is the

point where we use the countability assumption in HB 1). &

Axiomatizations of hyperbolic-type compactifications occur for example in

Beardon [5] and Kaimanovich [23]. Axioms for contractive compactifications, or

so-called convergence group actions, have also been considered by several people,

including Gehring and Martin [17], Gromov [18], Woess [46], and Bowditch [7].

From a less hyperbolic point of view, various notions of contractive boundaries were

introduced by Furstenberg around 1970.

3. Individual Isometries

Let X be a complete metric space andX an Isom(X )-bordification satisfying CB 1

and CB 2. The following proposition provides a classification of the elements in

Isom ðX; d Þ into elliptic isometries (bounded orbits), parabolic isometries (unbounded

orbits and one fixed point) and hyperbolic isometries (unbounded orbits and two

fixed points):

PROPOSITION 1. Let g be an unbounded isometry. Then there exist two not

necessarily distinct points x� 2 @X and nk such that gnkx ! xþ and gnkx ! x for

all x. Furthermore the set of fixed points inX of g consists exactly of the pointsðsÞ x�.


Proof. The convergence statement is asserted by CB 2. From continuity and CB 1

we have

gðx�Þ ¼ gð limk!1

g�nkxÞ ¼ limk!1

g�nkðgxÞ ¼ x�:

Moreover, CB 1 also implies that g cannot fix any other point. &

The following discussion will not be needed later and we shall therefore be rather

brief.

First note that in general, it is not true that for an unbounded isometry g the whole

sequence gnx converges when n ! 1 as is illustrated by Edelstein’s example ([24,

p. 1453]) together with a warped product giving an infinite-dimensional hyperbolic

space. However, in the case g has two fixed points in @X, then, at least under some

assumptions, one can say more:

For example, one can embed the complete metric space X isometrically in CðX Þ=R

via

F : x 7! ½dðx; �Þ�

as noted by Kuratowski and others) and let X ¼ FðXÞ. For CAT(0)-spaces this givesthe usual bordification, see [8, II.8.13]. Assume now that thisX satisfies HB 1 and

HB 2 (which is the case for CAT(-1)-spaces). Say that g fixes x 2 @X, then choosing

the representative b for which ½b� ¼ x and bðx0Þ ¼ 0, we have that

bðgyÞ ¼ bðyÞ þ Tg

for any y 2 X and some constant Tg. Hence, bðgnx0Þ ¼ nTg.

We now claim that if g fixes also another point, then jTgj > 0. Indeed, for gni ! ½b�

we get from Tg ¼ 0 that for any given m and e > 0 that

ðgnix0jgmx0Þ5 dðgmx0; x0Þ e

for all large i. This makes it impossible for any subsequence to converge to anything

else than x in view of HB 1.In general it holds that

Ag :¼ limn!1

1

ndðgnx0; x0Þ5 jTgj

and for CAT(0)-spaces one has in addition that Ag ¼ infx dðgx; xÞ. Assume that

Ag > 0, then one can show that gnx0 ! x and bðgnx0Þ ¼ Agn, cf. [24]. To sum

up we have:

THEOREM 5. Assume that X :¼ FðXÞ is a hyperbolic bordification and let g be an

isometry of X. Then one of the following occurs:

ð1Þ g has bounded orbits;

ð2Þ Ag ¼ 0, but orbits are unbounded and there is a unique fixed point in X and which is

the unique accumulation point @X;


ð3Þ Ag > 0, the forward and the backward orbits converge respectively to two distinct

points @X and these constitute the fixed point set of g.

4. Groups Generated by Two Unbounded Isometries

Assume throughout that X is an IsomðX Þ-bordification X and assume that the

conditions CB 1 and CB 2 hold.

PROPOSITION 2. Assume g and h are two unbounded isometries and that the group

generated by these two elements does not contain a non-Abelian free semigroup. Then

FixðgÞ ¼ FixðhÞ.

Proof. If j@Xj4 2; then any unbounded isometry fixes every point of @X. Assume

now that @X consists at least three points. From CB 2 we have g�nkx0 ! x� andh�nix0 ! Z�; for some, not necessarily all disjoint, boundary points. Suppose nowthat Fix(g) is not equal to FixðhÞ, so say xþ 6¼ Z. In the case Z ¼ Zþ ¼ x, wereplace h by hgh1. (The point is that this element will have forward limit point hxþ

which is neither equal to Z nor equal to xþ; the backward limit point has to be Z asbefore.)

Therefore we can assume that xþ is different from Zþ and Z, and that x isdifferent from Zþ. As X is Hausdorff we may find pairwise disjoint neighbor-

hoods U;V and W around xþ; Zþ and fZ; xg. From CB 1 there are m; n > 0 such

that

gnðWcÞ � U

hmðWcÞ � V:

These two elements generate a free non-Abelian semigroup, see for example Propo-

sition VII.2 in [21]. Hence, Fix(g) must be equal to Fix(h). &

PROPOSITION 3. Let g and h be two unbounded isometries. If their fixed point sets

are disjoint, then the group generated by g and h has a non-Abelian free subgroup.

Proof. First note that we may assume that the cardinality of @X is at least three

(hence infinite). Since X is Hausdorff we may find disjoint open sets X1 containing

FixðgÞ and X2 containing FixðhÞ. The statement now follows from CB 1 and the

standard table-tennis argument (Klein’s criterion) as in [21, Prop. II.24]. &

PROPOSITION 4. Assume in addition that CB 3 holds. Suppose g and h are

unbounded iosmetries of X and which generate a group which acts metrically properly

on X. If the fixed points sets of g and h are not equal, then the fixed point sets are in fact

disjoint.

Proof. This proof is a modification of a nice argument of Gehring and Martin

[17]. Both fixed point sets are nonemtpy by Proposition 1. In the case both g and h

have one fixed point then the statement is trivial. Therefore we now assume that

FixðhÞ ¼ fxþ; xg; ðh�n jx0 ! x�Þ and x 2 FixðgÞ: We need to show that xþ is fixed


also by g and we may therefore assume that @X contains at least three (hence infinite

number of) points. Choose neighborhoods U;Uþ in X of x and xþ, respectively,so that

hU \Uþ ¼ ;; ð1Þ

and

E :¼X n ðUþ [UÞ 6¼ ;;

which is possible because h is a homeomorphism, x� are fixed points of h andX is

Hausdorff. Since hnj contracts toward x and g is a homeomorphism fixing x,we have that

ghnj ðEÞ � U n fxg

for every large j. Because of (1) we can find a k ¼ kð j Þ such that

hkð j ÞghnjE \ E 6¼ ;: ð2Þ

Now let gj ¼ hkð j Þghnj and note that

gix¼ x and lim

j!1gjx

þ¼ xþ; ð3Þ

since kð j Þ ! 1 as j ! 1 (g is a homeomorphism) and ghnjxþ ¼ gxþ:We assert that gj is bounded. Suppose not, then by CB 3 (applied twice) and (3),

there is a subsequence ni such that g�1nix0 ! Z� 2 @X; where fn�g ¼ fx�g: But from

CB 1 we should then have that gniE � Uþ or U for all large i, which contradicts

(2). Hence, gj is bounded and by the properness assumption, gj ¼ gi for some distinct

i and j. Therefore hk ¼ ghlg1 for some nonzero intergers k and l.

We claim that it not follows that gxþ ¼ xþ. Indeed, applying the obtained equalityto gxþ we have

hkðgxþÞ ¼ ghlg1gxþ ¼ gxþ:

As hk is unbounded it can have at most two fixed points, namely the same as h, that

is xþ and x. So we have that gxþ is either xþ or x; but the latter is impossiblebecause gx ¼ x and g is bijective. &

5. Main Results: Unbounded Cases

Let X be a complete metric space and X a contractive bordification. For a group Gacting by isometry on X, we denote by LðGÞ the limit set of G, that is, the set of accu-mulation points in @X of an orbit Gx. The set LðGÞ is independent of which orbit weconsider by CB 1, and by continuity it is G-invariant. We call an action of G onX elementary with respect to X if the limit set LðGÞ consists of 0, 1 or 2 points. Ifthe orbit is bounded, then the action is said to be bounded. An action of G is calledquasi-parabolic with respect to X if G fixes exactly one point of the boundary @X:

THEOREM 6. Let G be a group in A. Assume that G acts on X such that the elements

of S are unbounded. Then, either the action is quasi-parabolic, or, G fixes two fixed

points of @X and the action is elementary.


Proof. Inductively using Propositions 1, 2 and that the NoFS2-graph is con-

nected, we have that the (nonempty) fixed point sets of each element in S coincide.

Hence, FixðGÞ, which equals this common set by CB 1, has cardinality 1 or 2. Nowassume the fixed point set contains two points Z and z, and that gnx0 ! x 2 @X. Then

CB 3 guarantees that there is a subsequence of g1n converging to Z in view of the factthat gnZ ¼ Z. Now CB 1 implies that x ¼ z. &

THEOREM 7. Let G be a group in B. Assume thatX also satisfies CB 3 and that Gacts on X such that the elements of S are unbounded. Suppose that the group generated

by any pair of elements of S acts metrically properly. Then the G-action is elementary

and fixes one or two points in @X.

Proof. Inductively using Propositions 1, 3, 4 and that the NoF2-graph is con-

nected, we have that the (nonempty) fixed point sets of each element in S coincide.

Hence, FixðGÞ, which equals this common set by CB 1, has cardinality 1 or 2. Finally,in view of Propositions 1 and 4, we have that the limit set must equal the fixed point

set using CB 3 similarly to the previous proof. &

If G is a finite extension of a group in A or B, then the above conclusions aboutelementariness of the action also holds for G because limit sets, if finite, can onlyconsist of at most two points for contractive boundaries.

Note that if two elements s and r are conjugate then both are either bounded or

unbounded as can be seen from the following simple calculation:

dðx; rnxÞ ¼ dðx; ðgsg1ÞnxÞ ¼ dðx; gsng1xÞ ¼ dðg1x; sng1xÞ:

6. Main Results: Bounded Cases

As usual we denote by X a complete metric space with a contractive bordificationX .

PROPOSITION 5. Assume that X is sequentially compact and let G be a group of

isometries of X. Suppose that every element of G is bounded. Then either the G-orbit isbounded or LðGÞ ¼ FixðGÞ � @X consists of exactly one point.

Proof. We may suppose that the orbit is unbounded and that G does not fix apoint in @X. By compactness we can find a sequence gn such that gnx0 ! xþ 2 @X

and g1n x0 ! x 2 @X. Moreover, if xþ ¼ x then as G does not fix this point thereis an element p 2 G such that pgnx0 ! pxþ 6¼ xþ and ð pgnÞ

1x0 ¼ g1n px0 ! x. Wemay hence assume that xþ 6¼ x. By CB 1, for every large N, gN is an unboundedisometry, cf. [26, Lemma 3]. This is a contradiction. &

We do not know if it can happen that the action is unbounded but the limit set is

empty in the case X is not locally compact.


A metric space (Y; d ) is called uniformly convex if it is geodesic and there is a

strictly decreasing continuous function g on [0,1] with gð1Þ ¼ 0, such that for any

x; y;w 2 Y and midpoint z of x and y,

dðz;wÞ

R4 g

dðx; yÞ

2R

� �;

where R :¼ maxfdðx;wÞ; dð y;wÞg. Cartan–Hadamard manifolds and more generally

CAT(0)-spaces, as well as Lp-spaces for 1 < p < 1 are standard examples of

uniformly convex spaces. A circumcenter of a bounded set D is a point x 2 Y such

that BrðxÞ � D and the radius r is minimal, that is

r ¼ inffR : 9y;BRðyÞ � Dg:

The following lemma is well known, but we include the proof for completeness and

in lack of a reference.

LEMMA 1. Let D be any bounded subset of a uniformly convex complete metric space

ðY; d Þ. Then there exists a unique circumcenter of D.

Proof. We may assume that D contains at least two points so that

r :¼ inffR : 9y;D � BRðyÞg > 0:

To prove the existence, let fxng be a sequence of points such that D � Brn ðxnÞ and rnconverges to r. We need to show that fxng is a Cauchy sequence. Let e > 0,d ¼ maxfrn; rmg=r, and z ¼ zn;m be the midpoint of xn and xm. Then there is a point

w ¼ wn;m such that dðz;wÞ5 r=d:By uniform convexity we have

dðz;wÞ4 gdðxn; xmÞ

2maxfrn; rmg

� �maxfrn; rmg < g

dðxn; xmÞ

2t

� �dr:

Hence

gdðxn; xmÞ

2r

� �>1

d2:

There is a large number M such that d is so close to 1 so that

dðxn; xmÞ < 2rg1ðd2Þ < e

for all n;m5M. By completeness xn converges to a point which is a circumcenter.

Note that because fxng was arbitrary it is clear that this circumcenter is unique. &

In view of the above two statements we obtain the following corollary which

generalizes a special case of a fixed point result in [40]:


COROLLARY 4. Assume in addition that X is uniformly convex and X sequentially

compact. If each element of G has a fixed point in X, then G has a fixed point in X .

Proof. By Proposition 5 we may assume that Gx0 is a bounded set. Since the mapassigning to a bounded set its unique circumcenter (Lemma 1) commutes with isometries

and because GðGx0Þ ¼ Gx0 we have that G must fix the circumcenter of this orbit. &

Let us now record a statement from the theory of locally compact hyperbolic

spaces which carries over:

PROPOSITION 6. The stabilizer of three distinct points in @X is bounded.

Proof. Let G be the group which pointwise fixes the three points in question x1,x2, and x3. It has finite index in the stabilizer. Suppose that G is unbounded, so thereis a sequence gn ! 1 and gnxi ¼ xi for every n and i. Then CB 3 implies that there isa subsequence nk such that gnkx0 ! x1 ðor g1nk x0 ! x1Þ. Applying CB 3 to thissubsequence we get a finer subsequence n1 for which g

en1x0 ! x1 and ge

n1x0 ! x2, for

e ¼ 1 or 1. As this sequence also fixes x3 we have a contradiction to CB 1. &

In the theory of convergence groups (see [7] and [17]) one deduces that G actsproperly on triples of points in the compact space @X.

A group is boundedly generated by a finite set of generators S if there is a constant v

such that every element g 2 G can be written as a product g ¼ rk11 rk22 . . . rkmm where ri

are some elements in S, ki integers and m4 v.

Note that the following lemma is related to an idea in [40, Theorem 2.6]:

LEMMA 2. Let G be a group boundedly generated by a finite set S. If G acts on a

metric space such that every element in S is bounded then every G-orbit is bounded.Proof. Let ri be a sequence of generators, 14 i4n: Denote by B1 a bounded

subset, e.g. fx0g: As r1 is bounded, we have that also

B2 :¼[n2Z

rn1B1

is bounded. By finite induction ð14 i4nÞ we can conclude that[

ðk1;...;kmÞ2Zm

r kmm � � � r k22 r k11 B1

is bounded. As the number of sequences ri; 14 i4n of elements in S is finite, we cancontinue the induction to obtain that GB is bounded for any bounded set B: &

Now putting together this last lemma with Theorem 6, the remark in the end of

Section 4 and Lemma 1, we obtain Theorem 3 and the following more general result:

THEOREM 8. Assume that G ¼ hSi is a group in A such that in addition G is

boundedly generated by S consisting of a finite number of elements all of which are


conjugate. Whenever G acts by isometry on a complete metric space X with a

contractive bordification, then either the orbit is bounded or there is a point in @X fixed

by all of G:

7. Examples of Bordifications

Here follows a list of hyperbolic IsomðXÞ-bordifications:

– the one-point bordification of any complete metric space. Trivial;

– the bordification construction of Floyd (or the conformal boundary in the

terminology of [18]) of geodesic complete metric spaces. The proof is essentially

Lemma 1 in [26], see also [25];

– the hyperbolic boundary of Gromov’s d-hyperbolic spaces (e.g., CAT(-1)-spaces). This is almost immediate from the definition (see [16]), but it is also

in fact a special case of the previous example;

– the bordification with the space of ends introduced by Freudenthal. This is

simple;

– the usual compactification of proper visibility spaces (first introduced by

Eberlein and O’Neill, see [8]). (A variant is the concept of a locally visible

CAT(0)-space, whose definition immediately verifies (HB 1). It seems that

(HB 2) is not so clear in the nonproper case;

– the ordinary closure of a strictly convex bounded domain in Rn with Hilbert’s

metric. This is due to Beardon [5] and [29, Section 5]. Note that here the exten-

sion property of isometries is not automatic.

Note that ifX is metrizable then the countability condition in HB 1 can indeed be

fulfilled. The same is true for the one point bordification and for the ray boundary

of a CAT(-1)-space. In the latter case, given a geodesic ray g, we may choose acountable number of shadows of balls centered on g constituting a fundamentalsystem of neighborhoods of the boundary point that g defines.

8. Examples of Groups

Recall the notion of P-graph defined in the introduction. Some of the following ser-

ies of examples are adaptions from [12].

8.1. TRIVIAL EXAMPLES

Nilpotent groups and finitely generated groups of subexponential growth do not

contain any non-Abelian free semigroup. Finitely generaed amenable groups, in

particular solvable groups, do not contain any non-Abelian free groups. The product

of two groups is generated by a set whose C-graph is connected. Coxeter groups

also belong to A since any pair of the (standard) generators generate a finite or

infinite dihedral group.


8.2. SL ðn;AÞ; n5 3

Let A be a commutative Z-algebra with 1, for example Z, a ring of S-integers, or a

field. An elementary matrix is an element xijðaÞ :¼ Iþ aeij of SLðn;AÞ where a 2 A

and eij is the matrix with 1 at the place ði; jÞ; i 6¼ j; and zero everywhere else. It is

straightforward to verify the following relations for any a; b 2 A:

½xijðaÞ; xklðbÞ� ¼ xilðabÞ

if i 6¼ l; j ¼ k, and

½xijðaÞ; xklðbÞ� ¼ 1

if i 6¼ l, j 6¼ k.

Let Eðn;AÞ denote the subgroup of SLðn;AÞ generated by all the elementary

matrices. The N-graph of the set of elementary matrices is connected provided n5 3.

To see this, let xijðaÞ and xklðbÞ be two elements, so i 6¼ j and k 6¼ l. If i 6¼ l, and j 6¼ k,

then they commute. If j ¼ k and i 6¼ l (or j 6¼ k and i ¼ l), then it also follow from the

above commutation rules that the two elements in question generate a 2-step

nilpotent group. Finally, if i ¼ l and j ¼ k, then there is an m different from i and

j (since n5 3). Now ½xijðaÞ; ximðaÞ� ¼ 1 and xklðbÞ ¼ xjiðbÞ together with ximðaÞ gener-

ate a nilpotent group as in the second case. This shows that Eðn;AÞ 2 A for n5 3.

Assume from now on that n5 3. If A is a field or the integers of a number field,

then Eðn;AÞ ¼ SLðn;AÞ, see [4]. More generally, if A satisfies Bass’ stable range

condition Sd (see [3]) and SK1ðAÞ is finite, then Eðn;AÞ has finite index in SLðn;AÞ

for n5 dþ 2, and hence the results in Section 5 apply. (Recall that SLð2;ZÞ acts

non-trivially on the hyperbolic plane.)

Assume that A is generated by a1; . . . ; am. One can then generate Eðn;AÞ with

all xijðakÞ and xijð1Þ (assuming n5 3). The group Eðn;AÞ contains the group of

monomial matrices which acts transitively by conjugation on the set of generators

fxijðakÞg for every fixed k. If the ais are units, then any two xijðakÞ and xijðalÞ are also

conjugate.

In [11] it is proved that when A is the ring of integers of a number field, then

SLnðAÞ is boundedly generated by elementary matrices. For A ¼ Z see also [1],

and for some interesting discussions on bounded generation see [40]. In these cases

A is isomorphic to Zn additively, which implies that every elementary matrix xijðaÞ

can be written as a product of xijðakÞ, 14 k4m, and xijð1Þ. Therefore such

SLðn;AÞ is boundedly generated by a finite set of elementary matrices.

In conclusion we have that, if A is a ring of integers of an algebraic number field

and ZAx ¼ A (the units generate A), then Theorems 3 and 8 apply to G ¼ SLðn;AÞ

for n5 3.

8.3. CHEVALLEY GROUPS OVER RINGS

Let A be a finitely generated commutative Z-algebra with 1 and F a reduced irredu-cible root system of rank greater than 3. The Steinberg group StðF;AÞ associated to F


with coefficients in A is generated by symbols xaðaÞ, where a 2 F and a 2 A subject to

the relations

(1) xaðaþ bÞ ¼ xaðaÞxaðbÞ

(2) ½xaðaÞ; xbðbÞ� ¼ xaþbðNa;babÞ; aþ b 6¼ 0;

where Na;b ¼ Nb;a if a; b; aþ b 2 F, and Nb;a ¼ 0 if a; b 2 F; aþ b =2F, [41].

PROPOSITION 7. The standard generating system S ¼ xaðaÞ : a 2 F; a 2 A n f0g has

a connected N-graph.

Proof. If Fþ is the set of positive roots relative to some ordering of F then thecorresponding subgroup is nilpotent, in particular the N-graph for the root sub-

groups xaðAÞ; a 2 Fþ is connected. Now say that two positive systems are adjacent if

they have a non-empty intersection. This gives a graph structure on the set of

positive systems. If we prove that this graph is connected this would give the desired

connectedness of the N-graph. The Weyl group acts transitively on the set of positive

roots [22], thus we need only connect Fþ to oFþ where o is a reflection in a simpleroot a of Fþ. But o fixes all simple roots but a and since F has rank at least 2, weconclude that Fþ and oFþ are adjacent. &

Given a reduced irreducible root system Fn of rank n and a commutative ring A

with 1, one can define a Chevalley group GðFn;AÞ, a group EðFn;AÞ generated by

the unipotents in GðFn;AÞ, and a Steinberg group StðFn;AÞ. There is a natural

homomorphism p : StðFn;AÞ ! GðFn;AÞ with image EðFn;AÞ, and one defines

K1ðFn;AÞ ¼ cokerp and K2ðFn;AÞ ¼ ker p; these constructions are all functorial inA. It follows from the above that the N-graph of EðFn;AÞ is connected. If

K1ðFn;AÞ is finite one could apply Theorems 6 and 7 to the Chevalley group

GðFn;AÞ. In particular this is the case when A is the ring of integers of a global field.

Results on bounded generation are obtained in [43]. If the Weyl group acts transi-

tively on the roots then xaðaÞ and xbðaÞ are conjugate as in the previous subsection.

This shows that there are some further situations where Theorems 3 and 8 apply.

Note, however, that in the case X is a complete separable CAT(-1)-space, the

papers [10] and [16] provide superrigidity results for general lattices in higher rank

simple groups.

8.4. BRAID GROUPS Bn; n5 5

The Artin presentation of the classical braid groups Bn is

sisj ¼ sjsi if ji jj5 2

sisiþ1si ¼ siþ1sisiþ1

for 14 i4 n 2. Hence the C-graph of fs1; s2; . . . ; sn1g is connected for n5 5.

(More generally, any Artin group, for which the associated standard Coxeter gener-

ating system has a connected C-graph, belongs to A.)


The generators si are all conjugate. Note that Bn is not boundedly generated.

Indeed, recall that the corresponding pure braid group Pn, which has finite index

in Bn, surjects onto P3 which is isomorphic to F2 � Z, and bounded generation

passes to finite index subgroup and to homomorphic images.

When a group has an element in the center which acts unboundedly, then the

arguments in Section 4 imply that the whole group must fix a boundary point. It

is well-known that this remark applies to the braid groups since the element

O2 ¼ ððs1s2 . . . sn1Þðs1s2 . . . sn2Þðs1 . . .Þðs1s2Þs1Þ2

is a central element of infinite order.

8.5. MAPPING CLASS GROUPS ModðSg;nÞ; g5 2

Let ModðSg;nÞ be the mapping class group of an oriented surface Sg;n of genus g

and n punctures.

It is known that if the genus is at least one, these groups are generated by a finite

number of Dehn twists around nonseparating simple closed curves and any two of

these curves are either disjoint or intersect transversally in one point. If disjoint, then

the corresponding two Dehn twists generate Z2. If 3g 3þ n5 3 then the C-graph

is connected for this generating set and so ModðSg;nÞ 2 A.Since any nonseparating simple closed curve can be moved to any other by

a homeomorphism, the elements of S are all conjugate. For more information,

see [9] or [12].

Farb, Lubotzky and Minsky proved in [13] that ModðSg;0Þ has finite index sub-

groups whose pro-p completion is non-p-analytic, and as a corollary they establish

that ModðSg;0Þ, for g5 1 is not boundedly generated by any finite set.

To see that ModðSg;nÞ, for g5 1, or g ¼ 0 and n5 4, is not boundedly generated

by any finite set of Dehn twists, we propose a quite different approach. Namely, it is

not a very deep fact that ModðSg;nÞ acts unboundedly by isometry (see [33, p.105,

p.124]) on the complex of curves. Moreover, it is immediate that each Dehn twist

fixes its twisting curve in the complex. Our claim now follows from Lemma 2.

In fact, taking advantage of the deeper aspect of [33], Bestvina and Fujiwara

recently managed to show that the space of nontrivial quasihomomorphisms

of ModðSg;nÞ as above, is infinite-dimensional. This property is incompatible with

bounded generation (this remark we learnt from M. Burger).

8.6. AUTOMORPHISM GROUPS OF FREE GROUPS

We follow [12] here. Let Fn be the free group of rank n5 3, with generators

x1; . . . ; xn. We first consider the index two subgroup SAutðFnÞ of AutðFnÞ consisting

of special automorphisms (as automorphism is special is the determinant of the

induced automorphism of Zn is equal to þ1). For i 6¼ j, let ri;j (respectively li;j) be


the automorphism which sends xi to xixj (respectively xjxi and fixes xk for k 6¼ i. The

elements ri;j and li;j generate SAutðFnÞ. If i; j, and k are distinct, then

½ri;j; rj;k� ¼ ri;k½li;j; lj;k� ¼ li;k :

:

It follows that the automorphisms ri :¼ ri;iþ1 and li :¼ li;iþ1 for 14 i4 n 1,

together with rn :¼ rn;1 and ln :¼ ln;1 generate SAutðFnÞ. Set S ¼ fr1; l1; . . . ; rn; lng.

The group SAutðFnÞ contains the alternating group on n letters and the automorph-

isms which send exactly two generators to their inverses. By conjugating by

appropriate such automorphisms one sees that the elements of S are all conjugate.

It is known, due to Sury, that AutðFnÞ and SAutðFnÞ are not boundedly generated.

Now let si denote ri or li. It is easily checked that si commutes with rj and lj if

24 ji jj4 n 2. Thus the C-graph of S is connected if n5 5 and most of our

results apply to SAutðFnÞ which belongs to A and therefore also to AutðFnÞ.

Acknowledgements

The authors would like like to thank Herbert Abels and the Universitat Bielefeld for

making this work possible. The first named author also gratefully acknowledges the

support of the FIM and thanks Marc Burger for some useful discussions related to

this paper.

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