Coxeter groups and buildings

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



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.




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.




1 Introduction

2 Abstract reflection groups and abstract buildingsGeometric reflection groupsChamber systemsBuildings

3 Their geometric realizationsCAT(0) metricvia chambersExamples




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





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.





- 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.





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.





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).





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.





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∼





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”?





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





The dotted black edges give the Cayley graph





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.)





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.





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 .





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.





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





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





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 .





An apartment is the image of a W -isometric embeddingW ↪→ Φ.




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)).





Next I want to describe a polyhedral metric on these spaceswhich is “nonpositively curved” (or CAT(0)) in sense ofcomparison triangles.





Coxeter zonotopesIf W is finite, then WS can be identified with the poset of facesof a convex polytope.

Permutohedron





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”).





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.





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.





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:





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.)





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.





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).





ConclusionRABs of arbitrary thickness exist.





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.)


Coxeter groups and buildings

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