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Coxeter; Groups; Buildings; Blow Up (Facets); Fullerenes; Carbon Nanotubes
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IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Coxeter groups and buildings
Mike Davis
OSU
Bowling GreenOctober 14, 2011
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
The theory of abstract reflection groups or Coxeter groups wasdeveloped by J. Tits around 1960. This is a much larger classof groups than the classical examples of geometric reflectiongps on spaces of constant curvature. Nevertheless,associated to any Coxeter gp W there is a contractible space Σon which W acts properly and cocompactly as a groupgenerated by reflections. My main goal today is to describe Σ.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Tits coined the term “Coxeter gp” in connection with the theoryof buildings. Actually, the bldgs of interest to algebraists all areassociated either to finite reflection gps on spheres (leading tospherical bldgs) or to Euclidean reflection gps (leading to affinebldgs).
In fact, ∃ bldgs associated to general Coxeter gps and each hasan associated contractible cx Λ, similar to Σ. These giveinteresting examples for geometric group theory.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
1 Introduction
2 Abstract reflection groups and abstract buildingsGeometric reflection groupsChamber systemsBuildings
3 Their geometric realizationsCAT(0) metricvia chambersExamples
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
Dihedral groups
r r´
-1 0 1
For m 6=∞, Dm can berepresented as the subgp ofO(2) which is generated byreflections across lines L, L′,making an angle of π/m.
rL
rL´
π/m
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
Some history
- In 1852 Mobius determined the finite subgroups of O(3)generated by isometric reflections on the 2-sphere.
- The fundamental domain for such a group on the 2-spherewas a spherical triangle with angles π
p , πq , πr , with p, q, rintegers ≥ 2.
- Since the sum of the angles is > π, we have 1p + 1
q + 1r > 1.
- For p ≥ q ≥ r , the only possibilities are: (p,2,2) for anyp ≥ 2 and (p,3,2) with p = 3, 4 or 5.
- Later work by Riemann and Schwarz showed there werediscrete gps of isometries of E2 or H2 generated byreflections across the edges of triangles with anglesintegral submultiples of π. Poincare and Klein: similarly forpolygons in H2.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
- Around 1850, Schlafli classified regular polytopes in Rn+1,n > 2. The symmetry group of such a polytope was a finitegp generated by reflections and as in Mobius’ case, theprojection of a fundamental domain to Sn was a sphericalsimplex with dihedral angles integral submultiples of π.
- Around 1890, Killing and E. Cartan classified complexsemisimple Lie algebras in terms of their root systems. In1925, Weyl showed the symmetry gp of such a root systemwas a finite reflection gp.
- These two lines were united by Coxeter in the 1930’s. Heclassified discrete groups reflection gps on Sn or En.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
Properties
Let K be a fundamental polytope for a geometric reflection gp.For Sn, K is a simplex. For En, K is a product of simplices. ForHn there are other possibilities, eg, a right-angled pentagon inH2 or a right-angled dodecahedron in H3.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
Conversely, given a convex polytope K in Sn, En or Hn st alldihedral angles have form π/integer, there is a discrete gpW generated by isometric reflections across the codim 1faces of K .Let S be the set of reflections across the codim 1 faces ofK . For s, t ∈ S, let m(s, t) be the order of st . Then Sgenerates W , the faces corresponding to s and t intersectin a codim 2 face iff m(s, t) 6=∞, and for s 6= t , the dihedralangle along that face is π/m(s, t).
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
Coxeter diagrams
Associated to (W ,S), there is a labeled graph Γ called its“Coxeter diagram.” Vert(Γ) := S.Connect distinct elements s, t by an edge iff m(s, t) 6= 2. Labelthe edge by m(s, t) if this is > 3 or =∞ and leave it unlabeled ifit is = 3.(W ,S) is irreducible if Γ is connected. The next slide showsCoxeter’s classification of irreducible spherical and cocompactEuclidean reflection gps.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
Spherical Diagrams Euclidean Diagrams
I (p) p2
H 53
H 54
F 44
ωA1∼
4 4B 2∼
6G2
∼
4F4
∼
E6
∼
E7
∼
E8
∼
E6
E7
E8
An
B 4n
Dn
nA∼
4B n∼
4 4Cn∼
Dn∼
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
Abstract reflection groups
QuestionGiven a gp W and a set S of involutions which generates it,when should (W ,S) be called an “abstract reflection gp”?
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
Two answersLet Cay(W ,S) be the Cayley graph (ie, its vertex set is Wand {w , v} spans an edge iff v = ws for some s ∈ S).First answer: for each s ∈ S, the fixed set of s separatesCay(W ,S).Second answer: W has a presentation of the form:
〈S | (st)m(s,t), where (s, t) ∈ S × S〉.
Here m(s, t) is any S × S symmetric matrix with 1s on thediagonal and off-diagonal elements integers ≥ 2 or thesymbol∞.
These two answers are equivalent!In either case, (W ,S) is called a Coxeter system
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
The dotted black edges give the Cayley graph
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
DefinitionA chamber system is a set Φ (of “chambers”) together with afamily of equivalence relations indexed by some set S
Two chambers c and d are s-adjacent if they are s-equivalentand not equal.
A gallery in Φ is a finite sequence (c0, . . . , cn) of adjacentchambers. It is type (s1, . . . , sn) if ci and ci−1 are si -adjacent.(So, its type is a word in S.)
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
DefinitionsΦ is connected if any two chambers can be connected by agallery.Given a subset T ⊂ S, a gallery of type (s1, . . . , sn) is aT-gallery if each si ∈ T .A residue of type T is a T -gallery connected component ofΦ.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
Suppose (W ,S) is a Coxeter system. For each T ⊂ S,WT := 〈T 〉.
ExampleW has the structure of a chamber system over S.
v ,w ∈W are s-equivalent ⇐⇒ they belong to the samecoset of Ws (the gp of order 2 generated by s).A gallery in W is an edge path in Cay(W ,S).(Here a “chamber” is a vertex of Cay(W ,S).)Its type is a word in S.A T -residue is a coset of WT in W .
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
T is a spherical subset of S if |WT | <∞.S := the poset of spherical subsets.
Reduced expressions
Given a word s = (s1, . . . , sn) in S, its value is w(s) is theelement w ∈W defined by w := s1 · · · sn.s is a reduced expression for w if l(w) = n.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
Buildings
Suppose (W ,S) is a Coxeter system.
DefinitionA chamber system Φ over S is a building of type (W ,S) if eachs-equivalence class has at least two elements and if it admits aW-valued distance function δ : Φ× Φ→W .This means that if s = (s1, . . . , sn) is a reduced expression forw and if c0,cn ∈ Φ, then there is a gallery of type s from c0 to cn⇐⇒ δ(c0, cn) = w
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
Example (The edge set of a tree)
Suppose W is the infinite dihedral group with generators s, t .Let T be a tree w/o terminal vertices. Any tree is bipartite. Callthe colors s, t .Let Φ = Edge(T ). An edge path in T gives a gallery. Thesequence of colors of the vertices it crosses gives a word in{s, t}. The word is a reduced expression if the path has nobacktracking.The W -distance between b,c ∈ Φ is defined as follows. Takethe edge path w/o backtracking from b to c. Its type gives aword in {s, t} and hence, an element w ∈W and δ(b, c) = w .
r r´
-1 0 1
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
DefinitionsIf W is
finite, Φ is a spherical bldg.a cocompact Euclidean reflectiongp, Φ is a Euclidean (or affine) bldg.a finite dihedral gp, Φ is ageneralized polygon.
rL
rL´
π/m
ExampleW itself is a bldg. W -distance δ : W ×W →W is defined byδ(v ,w) = v−1w .
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
Geometric reflection groupsChamber systemsBuildings
An apartment is the image of a W -isometric embeddingW ↪→ Φ.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
Geometric realizations
Suppose P is a poset. Its geometric realization |P| is thesimplicial cx with one simplex for each chain p0 < · · · < pk in P.
Recall T is a spherical subset of S if |WT | <∞ andS := poset of spherical subsets.WS := poset of spherical cosets =
∐W/WT . For a bldg Φ,
C:= poset of spherical residues in Φ.
K := |S| Σ := |WS| Λ := |C|.
Basic FactΛ (and in particular, Σ) is contractible (in fact, CAT(0)).
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
Next I want to describe a polyhedral metric on these spaceswhich is “nonpositively curved” (or CAT(0)) in sense ofcomparison triangles.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
Coxeter zonotopesIf W is finite, then WS can be identified with the poset of facesof a convex polytope.
Permutohedron
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
For W finite, theintersection of its Coxeterzonotope with afundamental simplicialcone is called a “Coxeterblock”
CW
BW
For a general W , the chamber K (= |S|) is subdivided intoCoxeter blocks. This defines a piecewise Euclidean metric on Λand Σ.
Theorem (Moussong)
This metric is CAT(0) (meaning that the space is simplyconnected and “nonpositively curved”).
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
CorollaryΛ (and Σ) are contractible.
Proof.Λ is CAT(0) =⇒ ∃ unique geodesic between any 2 points =⇒Λ is contractible.
Corollary (Bruhat-Tits Fixed Point Theorem)If a group G acts on a building C with a bounded orbit ofchambers, then it has a fixed point on Λ. Hence, G is containedin the stabilizer of a spherical residue.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
K := |S| Σ := |WS| Λ := |C|.
W is gp generated byreflections in the faces ofright-angled pentagon.
K is a pentagon.
Σ is the tessellation of H2
by pentagons
The CAT(0) metric is defined by declaring each quadrilateral tobe a Euclidean square.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
Another approach to defining the geometric realization of achamber cx or bldg is to declare each chamber to behomeomorphic to a certain space X and then declare howadjacent chambers are glued together.
Usually we take X = K (the geometric realization of S).Another possibility is X = ∆ a simplex with its {codim 1 faces}indexed by S. In that case we get the Coxeter complex.
Here are the details:
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
Alternate description
X a space, {Xs}s∈S a family of closed subspaces.S(x) := {s ∈ S | x ∈ Xs}.U(Φ,X ) := (Φ× X )/ ∼, where (c, x) ∼ (c′, x ′) ⇐⇒x = x ′ and c,c′ belong to the same S(x)-residue.U(Φ,X ) is formed by pasting together copies of X , one foreach chamber in Φ.Put Ks := |S≥{s}|. Then Λ = U(Φ,K ).
When Φ = W , two chambers (= copies of X ) meet along eachface of type Xs. For a general bldg there are two or more alongeach Xs. (Think: line versus tree.)
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
Right-angled bldgs (= RABs)
A Coxeter gp is right angled if each mst is = 2 or∞.A bldg is right-angled if its Coxeter gp is (ie all m(s, t) are 2 or∞).
Example (Product of trees)
If W = (D∞)n, then K is an n-cube. A bldg of type (W ,S) is aproduct of n trees.
Will show∀ right-angled (W ,S), ∃ RABs of type (W ,S) with chambertransitive automorphism gp and arbitrary thickness along facesof type s.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
Graph products of type (W ,S)
(W ,S) a right-angled Coxeter system.Let L1 be a graph with Vert(L1) = S andEdge(L1) = {{s, t} | mst 6=∞}.Let {Gs}s∈S be a family of finite gps. For each T ∈ S (i.e.,for each complete subgraph of L1), put
GT :=∏s∈T
Gs.
G = lim−→GT is the graph product of the Gs.
Put Φ = G. If g = gs1 · · · gsn ∈ G, put δ(1,g) = s1 · · · sn ∈Wand δ(h,g) = δ(1,h−1g). Then Φ is a RAB. The number ofchambers along a face of type s is Card(Gs).
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
ConclusionRABs of arbitrary thickness exist.
Mike Davis Coxeter groups and buildings
IntroductionAbstract reflection groups and abstract buildings
Their geometric realizations
CAT(0) metricvia chambersExamples
Kac-Moody gps and bldgs
Kac-Moody gps and bldgs
Suppose each m(s, t) ∈ {1,2,3,4,6,∞} and Fq is a finitefield.∃ a “Kac-Moody gp” of type (W ,S) over Fq. It has structureof a “(B,N) pair” and hence, gives a bldg Φ with chambertransitive action of G.q + 1 chambers meet along each face of type s.G acts properly on Λ× Λ (where Λ is the geometricrealization of Φ.)
Mike Davis Coxeter groups and buildings