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Coxeter; Groups; Buildings; Blow Up (Facets); Fullerenes; Carbon Nanotubes

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 on which W acts properly and cocompactly as a groupgenerated by reflections. My main goal today is to describe .

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 , similar to . 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 pi/m.

rL

rL

/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 pip ,

piq ,

pir , with p, q, r

integers 2.- Since the sum of the angles is > pi, we have 1p +

1q +

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 pi. 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 pi.

- 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 1930s. 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 pi/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 pi/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 called itsCoxeter diagram. Vert() := 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 is connected. The next slide showsCoxeters 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

A1

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 (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 is a finite sequence (c0, . . . , cn) of adjacentchambers. It is type (s1, . . . , sn) if ci and ci1 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

Definitions 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 of.

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 |

IntroductionAbstract reflection groups and abstract buildings

Their geometric realizations

Geometric reflection groupsChamber systemsBuildings

Buildings

Suppose (W ,S) is a Coxeter system.

DefinitionA chamber system o