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Home , Coxeter group, Reflection group, Triangle Group

GROUPS AND COXETER GROUPS

by

Kouver Bingham

A Senior Honors Thesis Submitted to the Faculty of The University of Utah In Partial Fulfillment of the Requirements for the

Honors Degree in Bachelor of Science

In

Department of Mathematics

Approved:

Mladen Bestviva Dr. Peter Trapa Supervisor Chair, Department of Mathematics

(

Jr^FeraSndo Guevara Vasquez Dr. Sylvia D. Torti Department Honors Advisor Dean, Honors College

July 2014

ABSTRACT

In this paper we give a survey of the theory of Coxeter Groups and Reflection groups. This survey will give an undergraduate reader a full picture of Coxeter theory, and will lean slightly heavily on the side of showing examples, although the course of discussion will be based on theory. We’ll begin in Chapter 1 with a discussion of its origins and basic examples. These examples will illustrate the importance and prevalence of Coxeter Groups in Mathematics. The first examples given are the <7„, and the group of isometries of the ^-dimensional . In Chapter 2 we’ll formulate a general notion of a in topological space X, and show that such a group is in fact a Coxeter

Group. In Chapter 3 we’ll introduce the Poincare Theorem for reflection groups which will vastly expand our understanding of reflection groups thereafter. We’ll also give some surprising examples of Coxeter Groups that section. Then, in Chapter 4 we’ll make a classification of irreducible Coxeter Groups, give a linear representation for an arbitrary

Coxeter Group, and use this complete the fact that all Coxeter Groups can be realized as reflection groups with Tit’s Theorem. I . TABLE OF CONTENTS

ABSTRACT ii

1 INTRODUCTION 1

1.1 A FIRST EXAMPLE ...... 1

1.2 EXAMPLES WITH REGULAR PO L Y T O PE S...... 4

2 ABSTRACT REFLECTION GROUP 9

2.1 GENERAL N O T IO N ...... 9

2.2 PROPERTIES OF REFLECTION GROUPS...... 11

2.3 LABELING, DELETION, AND EXCHANGE ...... 16

2.4 COXETER GROUPS AND TERMINOLOGY...... 21

3 THE POINCARE THEOREM FOR REFLECTION GROUPS 23

3.1 THE T H E O R E M ...... 23

3.2 EXAM PLES...... 24

4 CLASSIFICATION OF COXETER GROUPS 31

4.1 THE LINEAR REPRESENTATION OF A 31

4.2 CLASSIFICATION OF COXETER GROUPS ...... 33

4.3 TITS T H E O R E M ...... 34

4.4 DISCUSSION...... 36

5 REFERENCES 37

iii 1

1. INTRODUCTION

For two semesters I have had the pleasure and opportunity of studying Coxeter Groups with Professor Mladen Bestvina. Coxeter groups are synonymous with Reflection groups and are an indispensable in and in . Reflection groups have rich historical roots and are crucial to classifying polygonal of surfaces. Although this project did not venture into this theory, finite reflection groups play a significant role in the classification of Lie Groups and Lie Algebras, another important in mathematics.

Coxeter groups are groups with very specific presentations, and encompass a large num­ ber of important groups. This semester in this project we showed that every Coxeter Group is a reflection group and vice versa. We then studied the Poincare Polyhedron theorem, which formulates a precise general criterion for an arbitrary polyhedron to have in to guarantee that it is the fundamental domain of a of isometries generated by the reflections of its faces, which can be presented as a Coxeter group. This theorem

determines whether it is possible to tessellate a space with a given polyhedron. The orbit

of this fundamental domain, under the action of a reflection group, is the of the

space.

1.1. A FIRST EXAMPLE

A first example in a first course in group theory is the symmetric group of n elements.

This is the set of permutations of n elements, or the set of bijective functions from the

set {1,2,...,«} to itself. This group is commonly denoted an, and turns out to be a group

generated by reflections (or transpositions). Well consider the example where an acts on

W1. The following disscusion of this section is influenced by Chapters 1 and 2 of [8] and

will follow its general discussion.

A reflection, acting on M", is an isometric (a function equal to its own in­

verse), which is identity on a flcM ". on acts by isometries on Wl by per­ 2 muting the canonical orthonormal basis {e\,e2 ,.. • •

{ 1 ,2,... ,n}, the corresponding transposition of cn acts on R” as the reflection with the hyperplane fixed points

H jj = {(xi,X2, ■ ■ • )Xn) £ |X; Xy}

Thus, it is easy to see that also acts on, and preserves, the following subsets of W1.

1. The affine hyperplane En~1 = {{x\,x2 ,...,x n) e W 1 |xi + X2 -\------h xn — 1}

2. The unit Sn~l — {(xi ,X2 , ■ - ■ ,x„) G M”|xj +x% H------l-x^ = 1}

3. The intersection En~l f)Sn_1 commonly denoted Sn~2.

Case n = 3: In this case there are three transpositions of 03, commonly denoted si = (1,2), s2 — (2,3), and^3 = (3,1). The notation (1,2) signifies the exchange of 1 and 2, and similar for the others. In this case E 2 is the plan ex + ^ + z = 1, normal to the vector (1,1,1) in M3.

The ’s intersection with the positive quadrant is seen in Figure 1.

To see the transpositions above as reflections we present E2 in Figure 2; 03 acts as reflections about the three lines seen that divide E2 into chambers, which look like pie slices in the figure. The basis vectors e\,e2 and e$, are seen as points of intersection with the circle in the figure, as well as the points on the plane v = ^(—e\ + 2^2 + 2^3), w =

!(2ei + 2e2 — £3), and r = \(2e\ — ^2 + £3)- The divisions of the circle that these lines create are called the chambers of the reflection group. The fundamental chamber is denoted

Co in figure 2. All possible locations that Co is sent to by the action of these transpositions, and all products of transpositions, produces all other chambers, as they are labeled in the figure. Note that 53, as it has been defined above, does not occur in the figure. This is because of the simple relation sj = S]S2Si = S2S\S2. 3

Figure 1: Intersection of the plane E2 with the postive quadrant of R3.

As we will see in all cases of a general reflection group, <73 acts simply and transitively on these chambers. The lines separating the chambers are commonly called the mirrors of the reflection group. This example shows how these chambers arise naturally in the geometry of the space, which the reflection group acts. This is a general attribute of all reflection groups. In the next section we will introduce some of these general notions of a reflection group.

Figure 2: The chambers of E2 on which 03 acts. 4

1.2. EXAMPLES WITH REGULAR

First we’ll cover some basic definitions in order to discuss and realize the symmetry groups of as a reflection groups, and show them to share some properties with symmetric groups.

Throughout this section we let V denote an affine of some dimension n > 1.

Definition 1.1: A convex polyhedron in V is the intersection of a finite number of closed affine half-spaces.

Definition 1.2: A in V is a convex polyhedron Q which has a non-empty interior.

Throughout this section we’ll denote the group of all isometries of V by which P is invariant as G(Q).

By choosing an origin in V, and let H be any affine hyperplane in V which does not contain the origin, denote H + the closed half-plane of V limited by H and containing the origin. Create a polytope Q containing the origin and define a family of affine

{Hs}seS such that

e = r K seS

Qcf]H+ j£j for some proper subset J C S

For brevity (this is merely an example section) we’ll not prove the following facts but refer the reader to [5] for proofs of these and a general discussion, and we’ll simply rely on intuition for the purposes of these examples.

Fact 1. The set {Hs}ses is finite and determined by Q. Hsf]Q is a polytope in Hs for each s e S.

Definition 1.3: The polytopes Hsf]Q for every s E S are called the faces or the (n — 1)- 5 facettes of Q. Similarly defined are the the {n — 2)~facettes, (n — 3)-facettes, e.t.c.

Fact 2. Q is the convex hull of its O-facettes, i.e. it’s vertices.

The 1-facettes of Q are it’s edges.

Definition 1.4: A flag in Q is a nested sequence Fq C F\ C . ..Fn-\ where each F\ is a fc-facette. The set of all flags of Q is denoted F(Q)

There is a nice connection between regular polytopes and the action of G(Q) on the set

F(Q), and it is the following.

Definition 1.5: A polytope Q is said to be regular if G(Q) acts simply transitively on the set F(Q) of Q, i.e. in this case the cardinality of G(Q) will match that of F(Q).

Example 1.1: Let {e\,e2 , ■ ■ ■, en.\ i} to be the standard basis vectors of K'l+1. The poly­ tope defined by the convex hull determined by {ei,e2 ,... ,en+\}, is called the standard , denoted A", and it is the poly tope

n+ 1 { (jci , X2, .. - ,xn+1) G Mw+11 Y Xj = 1 & Xi > 0 V /}. 1=1

Of course A'1 resides in En. A2 is seen in Figure 1. In each case, giving the vector ^ -(e i +

---- 1- €n+l) the assignment of an origin for En makes it possible to view En a vector space.

Now for each element w G cr„+i there corresponds a flag of A" with an edge between ew(]j

and ew(2) ar|d every other pair of vertices of A” for that matter. It is interesting (and pretty) for illustration to picture the vertices of A” projected onto a circle. These pictures for n from 1 to 20 are seen in Figure 3. These are called Petrie [14]. 6

Figure 3: Petrie polygons, showing all vertices of the standard simplices on a circle for n up to 20 [15].

Example 1.2: We now move to Rn, to an interesting example the Standard Cube in n-dimensions denoted Cubn and analyze its group. Cubn is defined as the convex hull of the 2'1 vertices at locations Y!i=\ eiei where £, £ {1 ,- 1}.

Proposition 1.1. G(Cubn) is generated by reflections o/M", and is equal to the semi-direct product C" xi on, where C2 denotes n copies of the of order order 2.

Proof In G(Cubn) there are clearly reflections about the n coordinate hyperplanes (i.e the planes spanned by any subset {e,}/e/ where / is a subset with n — 1 distinct elements from 7

{1,2,..., n}). These reflections each generate a cyclic group of order 2, namely C2. These generate the C] of G(Cubn), which is normal.

Now consider the following, we show that Cubn is regular. Let v be any vertex of the

Cubn. There are n vertices connected to v, which constitute a convex hull for a Standard

Simplex A”-1. The case where n = 3 is shown in Figure 4, and each Standard Simplex for every vertex is depicted having its own color. Note that by construction the flags begin­ ning at the vertex v are in bijection with the total number of flags on this simplex A”" 1 in

F (A""1). This is because to each edge of the Cubn emanating from v terminates at a vertex of A”-1 , for which there are n — 1 choices for a face (a 2-facette) of a given flag in Cubn, and likewise n — 1 edges (1-facettes) of A”-1, and so on. This can be done at every vertex of the Cubn. Since A"-1 is regular, there’s a bijection between G(An_1) and F(An_1). This shows there’s a bijection between F(Cubn) and the symmetry groups on the A”- 1’s. Now since G(AW_1) < G(Cubn) shows that Cubn is regular.

Figure 4: Cubj, with the simplices A2 corresponding to each vertex, given its own color. 8

We can now simply calculate the order of G(Cubn) by counting the flags, there are 2n vertices for which there are n emanating edges of n — 1 faces, and so on. Thus \G{Cubn) | =

2nn\. Furthermore, G(Cubn) acts transitively on the A" 1 ’s and contains the ’’permutations” of these simplices, this accounts for 2"n! group elements, and is therefore the total group.

The self evident semi-direct product is Cft x a„. As discuss, both and on are generated by reflections when viewed as acting on M”, therefore G(Cubn) is as well. □ 9

2. ABSTRACT REFLECTION GROUP

2.1. GENERAL NOTION

In this section well briefly invoke some notions in Topology to construct the full gen­ erality of a reflection group. This will allow us to generalize the examples of the previous section. Also, it will allow us to define the key features of a refection group, and give them their canonical presentation as a Coxeter Group. From here after, a topological space X will be assumed to be path-connected, locally path-connected, and Hausdorff.

Throughout this discussion it will be helpful to the reader to refer to the examples in section 1 to help navigate the general notions of this section for more fluidly. Illustrations, references, and examples will also be provided along the way.

Definition 2.1: A reflection is an involution r : X —> X with the following properties:

1. The set of fixed points of r, Fix(r) = Mr, separates X into two connected open sets,

which are interchanged by r. These are called the mirrors or walls of r.

2. Every point of Mr has an arbitrarily small connected open neighborhood U for each

point in Mr, such that U\Mr has two components interchanged by r.

Definition 2.2: A group T of homeomorphisms of X is said to be a reflection group

provided

1. (Rl) r is generated by reflections.

2. (R2) The collection of all mirrors Mr is a locally finite family in X.

3. (R3) If two reflections r, r' G T with r ^ r1, then dim(X) — dim(MrnM r<) = 2. (This

says that the ’’codimension” of Mr n Mri is 2). This means any path (j) : [0,1] X can

be approximated by arbitrarily closely by a map whose image misses M ^ M ^ . 10

As was seen in the example of the previous section, a chamber is the closure of a component of X \ \JreT{Mr}.

Lastly, we can fix an arbitrary chamber Q and call it the fundamental chamber. And denote the set V by the set of reflections v e T such that

(Mv\[jM r)f]Q ^0 r^v

In words, this set V is the set of reflections surrounding the fundamental chamber Q, as did the reflections about the lines in from the previous section, labeled in Figure 2, s\ and

S2 surround the fundamental chamber Co, and action by these reflections sent Co to each of the other chambers. We found that si and S2 generated the reflection group 03, since

03 = ,s'i ,s'2s 1 . It is common to call the set V the set of simple reflections.

Proposition 2.1. T acts on the set of mirrors {Mr}rep, and Vg,r G T, g(Mr) — M g~ 1.

Furthermore, distinct reflections have distinct mirrors.

Proof. This proposition is a simple matter of checking each of the claims. By definition grg~l G r and by computing grg~1(gMr) = g(Mr) thus grg~l fixes gMr. Thus, g(Mr) C

M g~i. Now Vx e Mgrg- 1, grg~l (x) — x r(gx) — x i.e. r fixes gx. And trivially since g~ 1 = g implies g(gx) = x and therefore x G g{Mr). So g(Mr) — Mgrg- 1, and F acts on the set of mirrors.

Lastly, distinct reflections have distinct mirrors, since if not, and Mg = Mg> for some g,g' G r, then Mgf\M gi = Mg = Mg>. That is their intersetion separates X into two con­ nected open sets. But by (R3) Mgf]Mg/ has codimension 2, contradicting that any path in

X can be approximated arbitrarily close by a path that misses Mgf]Mgi. □

Definition 2.3 For a reflection v EV, the panel of v in Q is the set Qv = Qf]Mv.

Note that for Vg G T g(Qv) is a panel of g(Q). 11

2.2. PROPERTIES OF REFLECTION GROUPS

In this section we explore the core properties of an abstract reflection group developing the goal of proving that Q is indeed a fundamental chamber for T, and to realize T as a Coxeter

Group and give it’s canonical presentation. We begin with the following lemma.

Lemma 2.2. The subgroup Ty C F generated by V acts transitively on chambers.

Proof. Let Q' be any chamber. By the path-connectivity of X there exists a path from any point in the interior of Q' , intQ', any point of intQ. Choose a path (j). By properties (R2) and

(R3) we may choose (j) to miss all pairwise intersections of mirrors, and to intersect mirrors finitely often between Q' and Q. This determines a sequence of chambers that (j) crosses

Q = Qo,Qi■ iQn — 0!• There is a point x where

Qi to conclude that there is some w G F such that v-1 (Q2) = w(Q), i.e. Q2 = vw(Q), thereby sending Q to Q2 by action of v G V. We may continue this process sending Q to

Qn = Q' □

Definition 2.4: A gallery is a sequence of chambers Q — Qo, Q\ ,..., Qn = Q! so that

any two consecutive chambers are adjacent, i.e. there is a point x G Qn \ f) Qn that belongs to no other chambers.

Theorem 2.3. V generates T.

Proof. Since by (Rl) T is generated by reflections, the theorem will follow if we are able

to show that every reflection r G T is conjugate into V by some g G Ty, since by (Rl) T is

generated by reflections. To do this choose a chamber Q' so that one of it’s panels is Mr.

This is possible since for any two points on opposite sides of Mr can be joined by a path

which misses intersections with Mr, a neighborhood of the intersection of this path and

Mr determines two such chambers. By Lemma 2.1 Fy C T acts transitively on chambers,

therefore there is a g G Tv such that g(Q) = Qr. By Remark 2, g_1rg is a reflection whose

mirror contains a panel in Q so g~lrg G V. This proves the theorem. □ 12

Now that we’ve developed that fact that V generates V, every element g G F can be written g = V\V2 ■■■v„ s.t. v, e V for each i. Thus, we’ve have motivated the following definition.

Definition 2.5: For any element g E T define the length of g, denoted 1(g), as the small­ est n such that g = vj V2 ■ ■ ■ vn for v* e V.

Also, it should be noted that the sequence Q,v\(Q), v\V2 (Q), ■ ■ ■ ,v\V2 • • -vn(Q) = g(Q) is a gallery, and 1(g) is length of the shortest gallery joining Q to g(Q).

We’ll need the following important Lemma to prove a few facts.

Lemma 2.4. Suppose that v, w E V and that gv(Q) and g(Q) are on opposite sides of Mw.

Then gv = wg.

Proof. Refer to Figure 3 throughout the proof for illustration. By assumption g(Q) and gv(Q) are on opposite sides of the mirror, therefore Q and v(Q) are on opposite sides of g~x(Mw) — Mg-l . They must also be on opposite sides of Mv. Using (R3) and considering a point x e 2Plv(2) that belongs to no other chambers than Q and v(<2),7898 we see that this is possible only if My — Mg~iwg. Since distinct reflections have distinct mirrors, it follows that v = 1 wg □ 13

Figure 5: Illustration for Lemma 2.3

We now have the following Lemma.

Lemma 2.5. Ifg(Q) = Q, then g = 1.

Proof. For such a g suppose that 1(g) = n > 1. Then in the gallery

j3,vi(e),v1v2(e),...,viv2---vn(e) =g(Q)

there is an i such that v\v2 - ■■ v,(<2) and viv2 ■ ■ ■ v;V;+ i (Q) are on opposite sides of Mv,. By

Lemma 2.3, (vi v2 • • • v/)vj+ 1 = Vi (vi • • • y;) = v2 • ■ ■ Vf so therefore Z(g) < n, contradicting

our assumption on/(g). □

Proposition 2.6. Distinct chambers are separated by at least one mirror.

Proof. By Lemma 2.2 V<2 ^ Q' there exists g so that g(Q) = Q'. If 1(g) = 1 then g 6

V and Q and Q' are separated by Mg. If 1(g) > 1 then since Ty — T g = viv2 ■ ■ • v„ for

v,: G V and the sequence vi (Q), v\v2(Q),..., viv2 • • • vn(Q) is a gallery of shortest length 14 separating Q and Q' by definition of 1(g). Q and Q' are separated by the sequence of mirrors MVl, v\(Mn vi v2 ■ ■ • vn-\ (MVn). □

Lemma 2.7. If g — v\v2 ■■■vn then 1(g) ^ n iff corresponding gallery

Q,n(Q), viv2(g), • • • ,viv2 • ■ ■ v„(G) = g(g) never crosses the same mirror twice.

Proof. This arguement is the same as in Lemma 2.5. If the aforementioned gallery does cross a mirror twice, then there will be some least j > 2 such that v\v2 ■ • • Vj(Q) and v\v2 ■ ■ ■ VjVj \-\(Q) are on opposite sides of the first such mirror that is crossed twice, call it M. By Proposition 2.1 M = M(ViVr..Vj)Vj i(VlV2...Vjyt- Now by assumption there ex­ ists some k > j + 2 so that the gallery crosses M the second time, i.e. vj ■ ■ ■ Vk(Q) and vi ■ ■ ■ VfcVfc+ 1 (<2) are on opposite sides ofM. By Lemma 2.4, (vi • • ■ VjVj+iVj+ 2 ■ ■ -Vk)vk+\ —

((v\V2 ■ ■ -Vj)Vj+ 1 (viv2 • • • vy)_1)(vi ■ ■ -VjVj+iVj+2 ■ ■ ■ vk)

= vi • ■ • vjVj+2 • • • Vk, and therefore 1(g) < n.

Theorem 2.8. Q is a fundamental domain for the action ofT. i.e

1. TQ — X

2. If x,y G Q and 3g G T such that g(x) = y, then x = y and g belongs to the subgroup

ofY generated by {v G V\x G Mv}

Proof. (1) follows from the fact that F acts transitively on the chambers and that the mirrors are nowhere dense in X.

To prove (2), again we write g — viv2 ■ ■ ■ vn for v;- G V, such that 1(g) — n. Now since Q and g(Q) are on opposite sides of the all the walls that the gallery Q, v\(Q), v\v2(2), •. •, v\v2 ■ ■ ■ vn(Q) crosses, we must have that y G Qf)g(Q)- By Proposition 2.1, these mirrors are a sequence of fixed sets MVl ,MV|V2V-i )^(viv2)v3(vlv2)-1 etc- Thus y, and hence x, are fixed by each v,; and therefore x = y, furthermore x G MVj for each v,. □ 15

Corollary 2.8 The action of T on X is properly discontinuous.

Proof. We need to show that for any x E X and any neighborhood U of x that intersects only finitely many chambers <2i, 2 2 , • ■ •, Qn, the set {g E r|g([/) ^ 0 } is finite. Let U be such a neighborhood. If g(U)f]U ^ 0 , then g(Qi) Qj for some i and j. Theorem 2.8 now implies the result, since it says there can only be one such g for a fixed i and j. □

We have completed our goal in showing that Q is a fundamental domain for T, and we consider some facts which follow from our findings.

Proposition 2.9. For any fixed chamber Qq, and for any other two distinct chambers (Q\ and Q'p the list of walls separating Qo from Q\ and Q\ yield distinct lists.

Proof. Denote g,g! E T such that g(Qo) = Q\ and gf(Qo) = Q\ and 1(g) = n and l(g') = n'. The lists separating Qq from Q\ and Q\ will be of the form {MVl,Mv^ v~ and

{Mvi ,My, v, ^-i,...} respectively. If these lists were the same then each element of one list

would have a corresponding equal element from the other list, i.e. ^(vr -v)v/+i(v1--vJ-)~1 =

M{v\-v,k)v,k+l fy'-.y;,) 1 f°r some j. k. But, this only happens when each j = k and each v, = v', this contradicts g ^ g'. ■ □

Proposition 2.10. The collections of chambers is locally finite.

Proof Suppose not, that there exists a neighborhood U C X so that the list of intersecting

chambers is infinite. Then fix such any two distinct Q and Q' in this list. By the local path

connectivity of X there exists a path 0 from a point x £ intQ to a point x' e intQ', and (j)

crosses each mirror that separates Q and Q', this list of mirrors separating them is unique by

Proposition 2.9 (and is finite). Since if there are infinitely many s intersecting U, there

are infinitely many unique lists made in this way, and therefore, there’d need be infinitely

many mirrors intersecting U. But this contradicts that the mirrors are locally finite. □ 16

2.3. LABELING, DELETION, AND EXCHANGE

There is a convenient way to think about a gallery’s correspondence to a word in V. What one can do, for a given chamber Q is label each panel Qv for v E V by v, and then label each of the r translates of Qv by v also. Then any gallery that from Q to another chamber Q! (not necessarily shortest) has a corresponding word in V that can be read off from the labels that a path crosses as it follows the given gallery from Q to Q'. In Figure 6 is a gallery from a chamber Q to Q!, the corresponding word can be read of by the path seen, cacdbdcbd, that is cacdbdcbd(Q) = Q'.

What is left to be shown is that by labeling will always be consistent, i.e. it is well defined.

Proposition 2.11. The labeling of the panels of all chambers of T defined by labeling a panel Qv by v, and all the T translates of Qv by v is well defined.

Figure 6: A gallery from Q to Q' and its corresponding word given by the labeling of panels.

Proof. This follows from Theorem 2.8, since it says that for any two distinct elements v,w EV, Qv and Qw are not T-translates of one another. □

Theorem 2.12. The Deletion Principle: If g — v\v2 ---vn and 1(g)

g = V\V2 ’ ' ' Vi ■ • • vj • ■ ■ vn.

Where the hats represent omission.

Proof. By Lemma 2.7 g has length n iff the corresponding gallery

Q,vi(Q),viv2(Q),. ■. ,viv2 ■ • • v„(Q) = g(Q) never crosses the same mirror twice. Since by assumption 1(g) < n, we’ll be done if we show that we get the same group element by simply deleting the v, and vj (where the gallery crosses such a mirror twice). Call this mirror M. So by construction, v\v2 ■ • • v;_\ (Q) and vi V2 ■ ■ • Vi(Q) are on opposite sides of M. Likewise for the chambers corresponding to j and j — 1. As they have been listed here, we’ll name these chambers 1, 2, 3, and 4 respectively, and they are shown in Figure 2.12. The path that the aforementioned gallery defines is shown passing in order between these chambers. We’ll show that by deleting v-, and vj we get a new path, which seen in the figure following that arc on the right, and the resultant path avoids passing M and still corresponds to the element g.

Figure 7: Illustration for Theorem 2.12. 18

The list of labels that the gallery crosses after passing from chamber 1 to 2 is v;,..., vj.

By Proposition 2.11, there is a label on chamber 1, corresponding to the label sending

chamber 2 to the chamber v\ ■ ■ ■ Vi+iVi+ziQ), a chamber on the path in the figure on the left.

Deleting vt and reflecting chamber 1 by this cooresponding label will send it to a chamber

on the opposite side of M, depicted in the picture on the path on the right. This process is

true for each chamber in the sequence of chambers from chamber 2 to 3. So by Proposition

2.11 we can write

vj+i = Vi---Vj{Q)

=> vi ■■■v1-_ivf+i •■■Vj-iiQ) = vi ■•■Vj(Q)

The chamber on the right is chamber 4, and the chamber on the left is the result of deleting

vi and Vy_i, from which the theorem follows. □

Theorem 2.13. The Exchange Principle: Let g e T with 1(g) — n, and suppose that g —

v\V2 ---vn — u\u2 ---un are two shortest word representatives of g. If v\ ^ u\, there is

another shortest word representative of g obtained from v i V2 ■ ■ ■ vn by putting u\ in front

and omitting one of the Vi’s.

g = U\V\ • • • Vj_iVj-)-i • •' V,j for some i > 1.

Proof By Lemma 2.7 the gallery corresponding to V1V2 • • • vn crosses MU]. Then there is

an i so that vj • • • v;_i (Q) and v\ ■ • • v((<2) are on opposite sides of MUl. By Lemma 2.4

u\v\ ■ ■ • Vf_ i = vi • --Vi, from which the claim follows. □

Theorem 2.14. Any shortest word representative of g can be transformed to any other by

substitutions of the form

vwvw • • • = wvwv • • •

where both sides have length m and (vm)m = 1. 19

Proof. We’ll proceed by induction on the word length /(g) = n. The statement is vacuous for n = 0,1. So suppose the statement is true for all words of length n — 1 > 1, for some n.

We prove n.

Let g — vi ■ • - vw = «!■■■ un be shortest word representatives for any element g. If vi = u\ or vn = un then we can cancel these letters and get a word of length n — 1 and we are done. So suppose this is not the case. By the Exchange principle, we can write g = u\viV2 • ■ ■ v,_iV;+i ■ ■ ■ vn, and this word has the same first letter as u\ ■ ■ ■ un so we can transform u\ ■ ■ ■ un to it by the induction hypothesis. Likewise, if i ^ n (namely the letter

deleted is not vn) then it also has the last letter of vi ■ • ■vn, and we can transform from it to

this word, making a transformation from u\ ■ ■ ■ un to vi ■ ■ ■ vn, as desired.

So suppose i = n, i.e. g = u\vi • • • v„_i, then again by the exchange principle we can

write g = V\U\V\V2 ■ ■ ■ Vj~\Vj+\ • • ■ v„_i, and again this has the same first letter as v\ ■ ■ ■ vn,

and if j ^ n — 1, it has the same last letter as u \V \ ■ ■ ■ vn- \ . So, again we are done unless

j — n — 1, in which case we repeat the process. We continue in this way until an omitted

letter is not the last one, or we are able to write g — v\u\ ■ ■ ■ (length n), and in this case we

make the transformations

Vl ’ ’ ■ Vyi -Y V\lt\ * • * '—) U\V\ * * * -y U l U2 * * " Uyi.

Where here the middle arrow is an allowed move by the theorem, giving the final transfor­

mation desired. □

Lastly, using the theorems of this section we give a presentation for a reflection group,

realizing them as Coxeter Groups (a Coxeter Group being any group with the presentation

of the following theorem.)

Theorem 2.15. A presentation for T is given by

< V\(vu)m(v,u^ = 1 foru,v G^>.

Where m(v, u) G {1,2,..., <»} is the order of the product (vu) G F. 20

Note that in this case, m(v, u) = 1 iff u = v, and m(v, u) = m(u, v) for all v,ue V. Lastly, relations of the form (v«)°° = 1 are discarded.

Proof. We already know that V generates F by Theorem 2.3. Also, by Theorem 2.14 we know that two word representatives of shortest length can be transformed to one another by using only the relations given. We need to show that any two word representatives of a group element can be transformed using the given relations. So, if we can show that any group element can be shortened using the relations given, we’ll be done, since we’d first need to shorten to shortest length then transform.

Suppose a group element u\u2 ---um is a word representative for the group element g which is of shortest length, and V\V2 ■ • • vn is word representative of g that is of shortest length. Then find k so that l(u\u,2 ••■«&) = k and l(u\u2 ■ ■ ■ w^fc+i) ^k+l. Then by the

Deletion Principle, we must have that l(u\u2 - ■■UjcU](+1) — k — 1 (we crossed one mirror twice). Thus, we have that

U\ ■ ■■Uj | ] = v\ ■ • • v*_ i .

Therefore, u\ ■ • • u;- = vi • • ■ are of shortest length, and therefore by Theorem 2.14 we can transform one to the other using only the given relations. This thereby transforms the original word U\U2 ■ ■ ■ um to

Vi ■ • ■ ] wf e + l ' ’ ' um-

We can then use the relation ul \ = 1 to shorten the word. This is what what was needed to be shown. □

It has now been shown that every reflection group can be realized as a Coxeter Group.

In Chapter 4 we’ll show that the converse is also true, that given a Coxeter presentation there is a linear action of such a group on an open convex cone in a vector space which makes this group a reflection group. However, first, in the next section, and in Chapter 3, we’ll develop a better understanding of Coxeter Groups and see more examples. We’ll then 21 proceed to the general classification in Chapter 4.

2.4. COXETER GROUPS AND TERMINOLOGY

Definition 2.6: For a finite set V, a Coxeter matrix of type V is a symmetric matrix

M = (m(v, u))V'Uav whose entries are either , or the symbol such that:

1. m(v,v) = 1 for all v G V,

2. m(v, u)> 2 for all distinct v,k GV.

Definition 2.7: A Coxeter Graph is a weighted graph that encodes the information of a corresponding Coxeter matrix M, denoted &M- It’s vertex set is V, and a pair of vertices

{v, u} are connected by an edge iff m(v, u) > 3 for m(y, u) G M. The weights of edges are defined as follows

1. When m(y, u) > 3 the edge {v, u} is given the weight m(v, u).

2. When m(v, u) = 3 the edge {v, u} is left blank.

Coxeter Graphs are given special names/symbols which we’ll introduce shortly.

Definition 2.8: A Coxeter System is a triplet where T is a group, V is a subset of r , and m :V x V ->{1,2,... ,00} a function such that

1. m(v, u) = 1 iff v = u for all v,mG7 .

2. m(v, u) = m(u, v) for all v, u G V.

3. Thas presentation: < V\{vu)m^ = 1 ,v,u G V >.

Example 2.1: We’ll show in the Chapter 3 that the presentation for the symmetric group

on+1 of Example 1.1 is

< si,s2,...,s n\s2j = \ fo r all Sj. (sisj)3 = 1 if \i- j\ = 1, and (stsj f = 1 if > 2 > . 22

This is also shown in [8], The element Sj corresponds to the permutation of{l,2,...,n+l} which transposes j and j + 1, i.e. sj = (j, j + 1).

Thus the corresponding Coxeter Matrix M has 1 ’s in the diagonal, 3’s in all off-diagonal entries and 2’s elsewhere.

The Coxeter Graph ^ is a segment with n vertices, thus n — 1 edges connecting them, all with weight 3 (thus left blank). The graph is shown in Figure 8, and is called An.

Figure 8: Coxeter Graph for <7n+i [13].

Example 2.2: The group on the n-dimensional cube G(Cubn), discussed in Example

1.2, is Cl xi <7n. From the previous example (and help from the next chapter) one works out that this group is isomorphic to the group defined by the presentation

< U\,U2 , ■ ■ ■ ,un-i,u n\(uiiij)3 = 1 for i,j .

This group is often called the signedpermutationgroup and it acts on the unit sphere in E” by permuting the coordinates (which corresponds to the on part of the semi-direct product), and changing the signs (corresponding to the C^ part of the product).

Its fundamental domain of G(Cubn) is the set

Q = {(xi,x2,...,x„) GEn|xw > ••• > xi >0}

And Q has n faces given by the equations xi — 0, and xj — Xj+\ for j — 2,3,..., n — 1.

Its Coxeter Graph is shown in Figure 9 and is often called Bn, or also Cn.

Figure 9: Coxeter Graph for G(Cubn) [13]. 23

3. THE POINCARE THEOREM FOR REFLECTION GROUPS

3.1. THE THEOREM

In this section we’ll focus on letting X be one of E”, or IF for n > 2 (where H” is

//.-dimensional 1). We’ll give the Poincare Polyhedron Theorem, which establishes some extraordinarily simple conditions on which a finite sided polytope in X must have in order to guarantee that the group generated by the reflections of its faces has this polytope as its fundamental domain.

Returning to the discussion in Section 1.2, we let Q be a convex polyhedron with a non­ empty interior (a polytope), which is the intersection of finitely many closed codimension

1 faces denoted {Hs}ses- As in Section 2, the set of reflections {rs|^ G 5} about a face

Hs, for s E S, generate the subgroup of isometries of X. With this, we can state Poincare’s

Theorem.

Theorem 3.1. Poincare Polyhedron Theorem (for reflection groups): If whenever two faces Hs and Ht of Q intersect in a codimension 2 face, the dihedral angle between Hs and

Ht has the form njm(s,t) for some integer m(s,t) > 2, then the following hold:

1. T ~ < rs,s G S\ r2 = 1, fo r 'is G S, (rsrt)m^s^ = 1 when m(s,t) is defined. >

2. Q is a fundamental domain for T.

3. Ifx,y G Q and 3g G T such that g(x) = y, then x = y and g belongs to the subgroup

o / r generated by the reflection {rsL.v' G Hs)

Proof. The proof can be found in [8] and [2], and will not be given here since it’s length

would double the length of this paper.

’For an introduction to HP see [9]. All examples given in this paper will deal with H 2, the hyperbolic plane, and theorems will work with general n. 24

Note that each of these 3 conclusions are already consequences of the axioms of a reflection group of Section 2.1. Thus, the Poincare Polyhedron Theorem is given to impose simple geometric conditions of a given polyhedron (namely the angles between the faces) in order to determine whether it is a fundamental domain of the group of isometries generated by the reflections of its faces. It also gives an expedient tool to writing down the Coxeter presentation, which by other means would often be considerably difficult.

3.2. EXAMPLES

We can begin exhibiting examples of the Poincare Polyhedron Theorem by starting with the simplest possible cases, i.e. those where the set S is as small as possible and increasing incrementally.

Example 3.1: If S = {5} i.e. one generator, then T = < sis2 = 1 >. This is C2, the cyclic group of order 2. The Coxeter graph is a single vertex.

Example 3.2: If S = {.?,?}, L has two generators, and T = < s,t\s2 — t2 — (st)n — 1 > for some integer n > 2. This is known as the of order 2n, commonly denoted

D2n. By Theorem 3.1, Z)2n acts as the set of reflections of an n-sided regular in the plane, and the product st is rotation of this polygon by 2%/n. The case where n = 6 is shown in the Figure, the angle between the mirrors of s and t in this case is 7t/6. The group symbol for £>2m is, depending on m, one of A2,B2,h(5), or /2(m) for m>l.

Figure 10: with fundamental domain of Dyi [12]. 25

Example 3.3 Triangle Groups'. When S = {x,y,z}, and p.q, r are integers larger that 2 or oo, then we can write

T = < x,y,z\x2 = y2 = z2 = (xy)p = (yz)q = (xz)r — 1 >

(if p,q or r is oo then the relation is deleted). By the Theorem 3.1, T is the group of isometries generated by reflections of a triangle with angles Ti/p, Tc/q, and %/r. We’ll denote such a triangle as T. From basic geometry, the sum of the angles of a triangle in E2 is equal to n, while the sum of the angles of a triangle on §2 is strictly larger than K, and in

H2 strictly less. This motivates defining

X-i — — 1 |----- 1 1—. 1 p q r

From which three cases follow:

If X — 1 then T is in E 2,

If X > 1 then T is in §2,

and if X < 1 then T is in H2.

There are only finitely many cases that satisfy the first two cases, and there are infinitely many that satisfy the last. The possibilities for the first two cases are listed in Table 1 and

2. The names of the trianlge tesselation that T induces is given, along with their group symbols.

(p, q, r) T Group Symbol (3, 3, 3) <4? (2 ,4 ,4 ) Square B2 (2 ,3 ,6) Bisected Hexagon G2

Table 1: Triangle Groups in E 2 26

(p, q, r) T Group Symbol (2 ,2, n > 2) (reducible) (2, 3,3) A3 (2, 3, 4) (or Cube) B \ (2, 3, 5) (or ) H3

Table 2: Triangle Groups of §2

( , , ) ( 2 . 3 , 6 ) 2 4 4 ______( 3 ^ 3 ) |

, . ; s . v r r x ' . . .■

A ' A i A W ^ W ¥

p - - / ' v - . \ a | j | | 1

! ...... ' " v : '

r ...... ■......

LAAAAAAAJ

p h y 0 4 3 c P 3 3 " v 9 T k rA f : A " / : - .

E P & W m J F k w k j £ k J ...... ^ ...... illH H H BH LU lM H M H W

bisected tetrakis trian gu la r tilin g

Figure 11: All possible triangle tesselations of E2.

For illustration, Figures 11, 12, 13, and 14 display pictures of the triangle tesselations of E2, 82, and H2. The type of tesselation is denoted by the group’s triplet (p,q,r), and is displayed above each picture. Also, note that each picture is checkered with two colors, this is to distinguish the orientation preserving from the orientation reversing isometries of r in each case. Orientation preserving isometries send like colors to like colors, while orientation reversing isometries send a color to it’s opposite. 27

(2,2,2) (2,2,3) (2,2,4)(2,2,5) (2,2, n)

f t ® © (2,3,4)(2,3,5)

Figure 12: All possible triangle tesselations of §2 (type (2,2,n) is ommited for obvious reasons). 28

Example right (2 p q>

(2 3 8) (2 3 « )

{2 4 f»)

(2 03 »)

Figure 13: Triangle tesselations of H2 of type (2,q,r).

| 3 so as)

Figure 14: Triangle tesselations of HI2 of general type (p,q.r). 29

The group symbols for the groups will be given in Chapter 4, as well as the Coxeter graphs for those given.

Example 3.4: This is probably the most surprising example in this paper, and is also found in [8], [2] , and [6],

Denote ±7 as the center of the group GL^iZ). Let PGL2(Z) be the

GL2(Z )/± I.

One proves that using basic linear algebra that PGL^iZ) is generated by the matrices

/ \ ( ^ / \ 0 11 -1 1 -1 0 A = , B = ’ C = 1 V1 °J 1 ° l J 1 ° V As is shown in when classifying the isometries of H2, as is done in [10], that PGL2(Z) acts la on H , in the following way. The matrix acts by 1 c d 7 az+b cz+d if ad — be > 0 az+b cz+d if ad — bc< 0

Thus, we can read off the action of A, B, and C on H 2. We get A : z >-* 1 /z, this is reflection in the unit circle, B: l—z, this is reflection in the line Re(z) = 1/2, lastly C : z •-> — Z-

We can draw these lines of reflection in the upper half plane, and they intersect making a triangle in H2 with angles 0, tc/2, tc/3. Miraculously, Poincare’s Theorem tells us that we can write a presentation for this group as follows.

PGL2(Z) A,B,C\A2 =B2 =C2 = (.AB)3 = (AC)2 = 1 >!!

We’ve already seen this group in Example 3.3. This is a . Figure 15 shows the corresponding tessellation of the upper-half plane. Figure 15: Tesselation of the upper half plane by (2,3,°°) triangles [3]. 31

4. CLASSIFICATION OF COXETER GROUPS

4.1. THE LINEAR REPRESENTATION OF A COXETER GROUP

In this section we follow [8] and [1J for our discussion. Let (T,V,m) be a Coxeter system with V finite.

Let E denote the vector space of functions from V to R and let {ev}vey be its canonical basis. Define the symmetric bilinear form B : E x E —> M by

B(ev,ew) — —cos(7t/m(v,w)) for all v, w G V

And in particular,

B(ev,ev) = 1 for all v G V

B(ev,ew) = 0 O m(v,w) = 2

B(ev,ew) = -1/2 <=> m(v,w) = 3

B(ev,ew) = -1 m(v,w) — °°

Now, for v G V define ov G End^E to be as follows

From which we can immediately check that

1. Gy(ev) = - 1

2. av has order 2.

3. codim{x G E\av(x) =x} = 1

4. Im( 1 — C7V) C Rev where Mev denotes the span of ev in E.

We also have that ov is a reflection of E. Which preserves B, namely for any x.y G E

B(ov(x),ow(y)) = B(x,y) 32

This says that ov is an orthogonal transformation.

We now have a candidate for a linear representation of an arbitrary Coxeter System

(r, V, m). In the classification of Coxeter Groups, we’ll therefore be concerned with whether such a representation is injective, or faithful.

To illustrate this representation in the general case to some degree of accuracy, we give the following example.

Example 4.1: Let V = {v. w}, the associated Coxeter Group presentation is the one from Example 3.2: T —< v,w\(vw)m<^v,w^ — 1 >. Set m — m(v,w). Writing (vw) — r so that vrv = wrw — r~l, then each element T can be written as rnvs where 6 G {0.1} and neZ.

When m > 3 this group is Cm xi C2.

Case 1: In the first case, we let 3 < m < oo. Then for all (x,y) E E2\{(0,0)}, we can calculate the form B

B(xev + yew,xev+yew) = (x — ycos(n/m))2 + (ysin(n/m))2 > 0 .

Thus, B is positive and non-degenerate. With respect to the scalar product B, then r is rotation of order m. This is the Dihedral group Dim- We have that in this case the map sending v i—>- c w is a faithful (injective) representation of T in the vector space

Case 2: In this case we let m = °°. And compute similarly as we did before, for any

(jc,y) e M2

B{xev +yew,xev+yew) = (.x - y ) 2.

Thus, the ker(B) is generated by the span of the vector ev + ew, and therefore therefore the representation is degenerate. Note that vectors in ker{B) are fixed by av and ow. Also, avaw has infinite order, since by setting v — ev — ew, then

ow (v ) = ev e w v .

Here again, the map sending v cv and w (->■ aw is a faithful representation of F, however, 33 as mentioned is degenerate.

We state the following theorem to invoke the generality of this example, although do not provide a proof since it requires much more discussion and development than what is here. In fact the following theorem is actually a corollary of other theorems concerning this material.

Theorem 4.1. Let (T,V,m) be a Coxeter system. IfY is finite, then the form B is positive and non-degenerate.

Proof. See [8], □

4.2. CLASSIFICATION OF COXETER GROUPS

Definition 4.1 A Coxeter system (r, V, m) is irreducible if the associated Coxeter graph is connected.

Definition 4.2 An irreducible Coxeter system (T, V,m) and the corresponding Coxeter graph are said to be Euclidean or Affine if the associated bilinear form B is positive and degenerate.

We saw examples of Affine Coxeter systems in the case of the triangle group examples in M2 and H2. The following theorem classifies all Affine Coxeter groups.

Theorem 4.2. Any Affine Coxeter graph appears in the following list.

Ey h

Figure 16: Affine Coxeter Groups [11]. 34

Proof. See [4]. I—1

It is left to classify finite Coxeter groups.

Theorem 4.3. (Coxeter; 1935) Any connected Coxeter graph for which the associated Cox­ eter group is finite appears in the following list.

•— •----- .,. ------group 1) 4 Bn *—•— ... -----•— • =» group Gg xs 2) Dn N — •----- ... — • => group C"-1 xi cfn of order 2n~l nl * (n>4) Es t----1-----j ----•— • =» group of order 51840 = 27 34 5

Er •— •---- j---- •-----•— • => group of order 2908040 = 2J0 34 5 7

E* *— *— -j---- •---- -— •— • => group of order 696729600 = 214 35 52 7

*4 •— •—• => solvable group of order 1152 = 27 32 6 . , ' G2 •— • =*• dihedral group C6 >1 C7 of order 12 5 ffs 4— i— -# =s> group A n x C2 of order 120 = 2s 3 5 _ 5 •— •-----•— • =>■ group of order 14400 = 26 3® 52

hip) (p=5 or p> T) =► dihedral group C >3 C2 of order 2p

Figure 17: Finite Coxeter Groups [8].

Proof. See [7] or [8], □

4.3. TITS THEOREM

We now conclude by showing the converse of the correspondence between reflection groups

and Coxeter groups. The goal will be to show that every Coxeter group can be realized as

a reflection group. This proved by a theorem known as Tit’s Theorem, and like the others 35 of this section, the proof can be a little involved. We will not give the proof here, but we will discuss the theorem to complete the picture we began seeking. We’ll continue with the notation introduced in Section 4.1.

We begin by considering the dual space of E, E* and define the Tit’s representation

cr* : r —> GL(E*) as the contragradient of the geometric representation, that is o*(w) is the transpose of the map a^w” 1) for all weT.

Next, we define the open simplicial cone

{x* G E*\x*(ev) > 0 fo r all v G V}.

Lastly, the Tit’s Cone is

Q = U W(C)- we r where w(C) denotes the image by o*(w) of the closure of C.

Theorem 4.4. Tit’s Theorem:

1. Let vi’F be such that w(C) f)C ^ 0. Then w = 1. In particular c* is injective with a

discrete image in GL(E*) and the same holds for a. The group W operates simply

transitively on the set {w(C)}wer-

2. The action ofT on Q is properly discontinuous, and £1 is a convex cone in E*.

3. Let (r, V, m) be an irreducible Coxeter sysetm. Then the Tit’s cone coincides with E* iff the group T is finite, it is an open half-space iff'Y is Affine, and its closure does not

contain any line in the other cases.

Proof. See [4], or [1]. □

Corollary 4.5. Let (T, V, m) be a Coxeter system and let B be the associated bilinear form

on E. Then T is finite iffB is positive, and non-degenerate. 36

4.4. DISCUSSION

This completes our survey, since we’ve completed all of our goals. We’ve shown that any Coxeter system (r,V,m) there is a faithful linear representation T —) GL„(R) .We’ve also given a complete classification of irreducible Coxeter Groups. We also have extended our intuition and understanding of Coxeter Groups with the development of Poincare’s

Theorem.

One consequence of Tit’s theorem not discussed is that any Coxeter Group is virtually torsion-free and residually finite. The first means that every Coxeter Group contains a finite index subgroup H such that the only element of finite order is the identity, and the second means that for every g G T, g ^ ec there is a homomorphism h : G —> Cl to a finite group

G' where h(g) ^ eGf. These two facts comprise what is known as Selberg’s Lemma.

Coxeter Group theory is a rich and diverse subject and there are countless other avenues that the study of them lead to. 37

5. REFERENCES

[1] Mladen Bestvina, Coxeter Groups, class notes (2014).

[2] , Poincare Polyhedron Theorem for Reflection Groups, class notes (2014).

[3] Never Ending Books, Dedekind 1877, 2014, [Online; accessed 20-July-2014].

[4] Nicolas Bourbaki, Algebres de lie, Springer, 1960.

[5] Arne Br0 ndsted, An Introduction to Convex Polytopes, Graduate Texts in Mathemat­

ics 90 (1983).

[6] Kenneth S Brown, Buildings, Springer, 1989.

[7] Michael Davis, The geometry and topology of coxeter groups, vol. 32, Princeton Uni­

versity Press, 2008.

[8] P. De La Harpe, An Invitation to Coxeter Groups, Group Theory From a Geometrical

Viewpoint, 1991.

[9] John Ratcliffe, Foundations of Hyperbolic Manifolds, vol. 149, Springer, 2006.

[10] John Stillwell, Geometry of surfaces, Springer, 1992.

[11] Wikipedia, Coxeter group — wikipedia, the free encyclopedia, 2014, [Online; ac­

cessed 28-July-2014],

[12] , Dihedral group — wikipedia, the free encyclopedia, 2014, [Online; accessed

20-July-2014],

[13] , Finite coxeter — wikipedia, the free encyclopedia, 2014, [Online; accessed

28-July-2014],

[14] , — wikipedia, the free encyclopedia, 2014, [Online; accessed

25-July-2014], 38

[15] , Simplex — wikipedia, the free encyclopedia, 2014, [Online; accessed 25-

July-2014]. I Name of Candidate: Kouver Bingham

Birth date: June 29, 1987

Birth place: Bountiful, Utah

Address: (219 Kelsey Avenue Apt. 2) Salt Lake City, Utah, 84111