The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.
Review
"This book is one of those that grows with the reader: A graduate student can learn many properties, details and examples of Coxeter groups, while an expert can read about aspects of recent results in the theory of Coxeter groups and use the book as a guide to the literature. I strongly recommend this book to anybody who has any interest in geometric group theory. Anybody who reads (parts of) this book with an open mind will get a lot out of it." ---Ralf Gramlich, Mathematical Reviews
" The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory." ― L'Enseignement Mathematique
"[An] excellent introduction to other, important aspects of the study of geometric and topological approaches to group theory. Davis's exposition gives a delightful treatment of infinite Coxeter groups that illustrates their continued utility to the field." ---John Meier, Bulletin of the AMS
Review
"This is a comprehensive―nearly encyclopedic―survey of results concerning Coxeter groups. No other book covers the more recent important results, many of which are due to Michael Davis himself. This is an excellent, thoughtful, and well-written book, and it should have a wide readership among pure mathematicians in geometry, topology, representation theory, and group theory." ―Graham A. Niblo, University of Southampton
"Davis's book is a significant addition to the mathematics literature and it provides an important access point for geometric group theory. Although the book is a focused research monograph, it does such a nice job of presenting important material that it will also serve as a reference for quite some time. In fact, for years to come mathematicians will be writing 'terminology and notation follow Davis' in the introductions to papers on the geometry and topology of infinite Coxeter groups." ―John Meier, Lafayette College
From the Inside Flap
"This is a comprehensive--nearly encyclopedic--survey of results concerning Coxeter groups. No other book covers the more recent important results, many of which are due to Michael Davis himself. This is an excellent, thoughtful, and well-written book, and it should have a wide readership among pure mathematicians in geometry, topology, representation theory, and group theory." --Graham A. Niblo, University of Southampton
"Davis's book is a significant addition to the mathematics literature and it provides an important access point for geometric group theory. Although the book is a focused research monograph, it does such a nice job of presenting important material that it will also serve as a reference for quite some time. In fact, for years to come mathematicians will be writing 'terminology and notation follow Davis' in the introductions to papers on the geometry and topology of infinite Coxeter groups." --John Meier, Lafayette College
From the Back Cover
"This is a comprehensive--nearly encyclopedic--survey of results concerning Coxeter groups. No other book covers the more recent important results, many of which are due to Michael Davis himself. This is an excellent, thoughtful, and well-written book, and it should have a wide readership among pure mathematicians in geometry, topology, representation theory, and group theory." --Graham A. Niblo, University of Southampton
"Davis's book is a significant addition to the mathematics literature and it provides an important access point for geometric group theory. Although the book is a focused research monograph, it does such a nice job of presenting important material that it will also serve as a reference for quite some time. In fact, for years to come mathematicians will be writing 'terminology and notation follow Davis' in the introductions to papers on the geometry and topology of infinite Coxeter groups." --John Meier, Lafayette College
About the Author
Michael W. Davis is professor of mathematics at Ohio State University.
Finite groups generated by orthogonal linear reflections on [R.sup. n ] play a decisive role in
the classification of Lie groups and Lie algebras;
the theory of algebraic groups, as well as, the theories of spherical buildings and finite groups of Lie type;
the classification of regular polytopes (see or Appendix B).
Finite reflection groups also play important roles in many other areas of mathematics, e.g., in the theory of quadratic forms and in singularity theory. We note that a finite reflection group acts isometrically on the unit sphere [S.sup. n -1] of [R.sup. n ].
There is a similar theory of discrete groups of isometries generated by affine reflections on Euclidean space [E.sup. n ]. When the action of such a Euclidean reflection group has compact orbit space it is called cocompact . The classification of cocompact Euclidean reflection groups is important in Lie theory, in the theory of lattices in [R.sup. n ] and in E. Cartan's theory of symmetric spaces. The classification of these groups and of the finite (spherical) reflection groups can be found in Coxeter's 1934 paper. We give this classification in Table 6.1 of Section 6.9 and its proof in Appendix C.
There are also examples of discrete groups generated by reflections on the other simply connected space of constant curvature, hyperbolic n -space, [H.sup. n ]. (See [257, 291] as well as Chapter 6 for the theory of hyperbolic reflection groups.)
The other symmetric spaces do not admit such isometry groups. The reason is that the fixed set of a reflection should be a submanifold of codimension one (because it must separate the space) and the other (irreducible) symmetric spaces do not have codimension-one, totally geodesic subspaces. Hence, they do not admit isometric reflections. Thus, any truly "geometric" reflection group must split as a product of spherical, Euclidean, and hyperbolic ones.
The theory of these geometric reflection groups is the topic of Chapter 6. Suppose W is a reflection group acting on [X.sup. n ] = [S.sup. n ], [E.sup. n ], or [H.sup. n ]. Let K be the closure of a connected component of the complement of the union of "hyperplanes" which are fixed by some reflection in W . There are several common features to all three cases:
K is geodesically convex polytope in [X.sup. n ].
K is a "strict" fundamental domain in the sense that it intersects each orbit in exactly one point (so, [X.sup. n ]/ W [congruent to] = K ).
If S is the set of reflections across the codimension-one faces of K , then each reflection in W is conjugate to an element of S (and hence, S generates W ).
Abstract Reflection Groups
The theory of abstract reflection groups is due to Tits. What is the appropriate notion of an "abstract reflection group"? At first approximation, one might consider pairs (W, S) , where W is a group and S is any set of involutions which generates W . This is obviously too broad a notion. Nevertheless, it is a step in the right direction. In Chapter 3, we shall call such a pair a "pre-Coxeter system." There are essentially two completely different definitions for a pre-Coxeter system to be an abstract reflection group.
The first focuses on the crucial feature that the fixed point set of a reflection should separate the ambient space. One version is that the fixed point set of each element of S separates the Cayley graph of (W, S) (defined in Section 2.1). In 3.2 we call (W, S) a reflection system if it satisfies this condition. Essentially, this is equivalent to any one of several well-known combinatorial conditions, e.g., the Deletion Condition or the Exchange Condition. The second definition is that (W, S) has a presentation of a certain form. Following Tits, a pre-Coxeter system with such a presentation is a "Coxeter system" and W a "Coxeter group." Remarkably, these two definitions are equivalent. This was basically proved in. Another proof can be extracted from the first part of Bourbaki. It is also proved as the main result (Theorem 3.3.4) of Chapter 3. The equivalence of these two definitions is the principal mechanism driving the combinatorial theory of Coxeter groups.
The details of the second definition go as follows. For each pair (s, t) [member of] S x S , let [m.sub.st] denote the order of st . The matrix ( [m.sub.st] ) is the Coxeter matrix of (W, S) ; it is a symmetric S x S **matrix with entries in N [union] {[infinity]}, 1's on the diagonal, and each off-diagonal entry > 1. Let **
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(W, S) is a Coxeter system if <S|R > is a presentation for W . It turns out that, given any S x S matrix ( [m.sub.st] ) as above, the group W defined by the presentation <S|R > gives a Coxeter system (W, S) . (This is Corollary 6.12.6 of Chapter 6.)
Geometrization of Abstract Reflection Groups
Can every Coxeter system (W, S) be realized as a group of automorphisms of an appropriate geometric object? One answer was provided by Tits: for any (W, S) , there is a faithful linear representation W [right arrow] GL(N, R) , with N = Card(S), so that
Each element of S is represented by a linear reflection across a codimension-one face of a simplicial cone ITLITL. (N.B. A "linear reflection" means a linear involution with fixed subspace of codimension one; however, no inner product is assumed and the involution is not required to be orthogonal.)
If w [member of] W and w [not equal to] 1, then w (int(ITLITL)) [intersection] int(ITLITL) = (here int(ITLITL) denotes the interior of ITLITL).
WC , the union of W -translates of ITLITL, is a convex cone.
W acts properly on the interior I of WC .
Let [C.sup.f] := I [intersection] ITLITL. Then [C.sup.f] is the union of all (open) faces of ITLITL which have finite stabilizers (including the face int(ITLITL)). Moreover, [C.sup.f] is a strict fundamental domain for W on I .
Proofs of the above facts can be found in Appendix D. Tits' result was extended by Vinberg, who showed that for many Coxeter systems there are representations of W on [R.sup. N ], with N S)and ITLITL a polyhedral cone which is not simplicial. However, the poset of faces with finite stabilizers is exactly the same in both cases: it is the opposite poset to the poset of subsets of S which generate finite subgroups of W . (These are the "spherical subsets" of Definition 7.1.1 in Chapter 7.) The existence of Tits' geometric representation has several important consequences. Here are two:
Any Coxeter group W is virtually torsion-free.
I (the interior of the Tits cone) is a model for [E.bar]W , the "universal space for proper W -actions" (defined in 2.3).
Tits gave a second geometrization of (W, S) : its "Coxeter complex" [XI]. This is a certain simplicial complex with W -action. There is a simplex [sigma] [subset] [XI] with dim [sigma] = Card( S ) - 1 such that (a) [sigma] is a strict fundamental domain and (b) the elements of S act as "reflections" across the codimension-one faces of [sigma]. When W is finite, [XI] is homeomorphic to unit sphere [S.sup. n -1] in the canonical representation, triangulated by translates of a fundamental simplex. When (W, S) arises from an irreducible cocompact reflection group on [E.sup. n ], [XI] [congruent to] [E.sup. n ]. It turns out that [XI] is contractible whenever W is infinite.
The realization of (W, S) as a reflection group on the interior I of the Tits cone is satisfactory for several reasons; however, it lacks two advantages enjoyed by the geometric examples on spaces of constant curvature:
The W -action on I is not cocompact (i.e., the strict fundamental domain [C.sup.f] is not compact).
There is no natural metric on I that is preserved by W . (However, in [200] McMullen makes effective use of a "Hilbert metric" on I .)
In general, the Coxeter complex also has a serious defect-the isotropy subgroups of the W -action need not be finite (so the W -action need not be proper). One of the major purposes of this book is to present an alternative geometrization for (W, S) which remedies these difficulties. This alternative is the cell complex [SIGMA], discussed below and in greater detail in Chapters 7 and 12 (and many other places throughout the book).
The Cell Complex [SIGMA]
Given a Coxeter system (W, S) , in Chapter 7 we construct a cell complex [SIGMA] with the following properties:
The 0-skeleton of [SIGMA] is W .
The 1-skeleton of [SIGMA] is Cay (W, S) , the Cayley graph of 2.1.
The 2-skeleton of [SIGMA] is a Cayley 2-complex (defined in 2.2) associated to the presentation <S|R **>. **
[SIGMA] has one W -orbit of cells for each spherical subset T [subset] S . The dimension of a cell in this orbit is Card( T ). In particular, if W is finite, [SIGMA] is a convex polytope.
W acts properly on [SIGMA].
W acts cocompactly on [SIGMA] and there is a strict fundamental domain K .
[SIGMA] is a model for [E.bar]W . In particular, it is contractible.
If (W, S) is the Coxeter system underlying a cocompact geometric reflection group on [X.sup. n ] = [E.sup. n ] or [H.sup. n ], then [SIGMA] is W -equivariantly homeomorphic to [X.sup. n ] and K is isomorphic to the fundamental polytope.
Moreover, the cell structure on [SIGMA] is dual to the cellulation of [X.sup. n ] by translates of the fundamental polytope.
The elements of S act as "reflections" across the "mirrors" of K . (In the geometric case where K is a polytope, a mirror is a codimension-one face.)
[SIGMA] embeds in I and there is a W -equivariant deformation retraction from I onto [SIGMA]. So [SIGMA] is the "cocompact core" of I .
There is a piecewise Euclidean metric on [SIGMA] (in which each cell is identified with a convex Euclidean polytope) so that W acts via isometries. This metric is CAT(0) in the sense of Gromov. (This gives an alternative proof that [SIGMA] is a model for [bar.E]W .)
The last property is the topic of Chapter 12 and Appendix I. In the case of "right-angled" Coxeter groups, this CAT(0) property was established by Gromov. ("Right angled" means that [m.sub.st] = 2 or [infinity] whenever s [not equal to] t.) Shortly after the appearance of, Moussong proved in his Ph.D. thesis that [SIGMA] is CAT(0) for any Coxeter system. The complexes [SIGMA] gave one of the first large class of examples of "CAT(0)-polyhedra" and showed that Coxeter groups are examples of "CAT(0)-groups." This is the reason why Coxeter groups are important in geometric group theory. Moussong's result also allowed him to find a simple characterization of when Coxeter groups are word hyperbolic in the sense of (Theorem 12.6.1).
Since W acts simply transitively on the vertex set of [SIGMA], any two vertices have isomorphic neighborhoods. We can take such a neighborhood to be the cone on a certain simplicial complex L , called the "link" of the vertex. (See Appendix A.6.) We also call L the "nerve" of (W, S) . It has one simplex for each nonempty spherical subset T [subset] S . (The dimension of the simplex is Card( T ) - 1.)If L is homeomorphic to [ S.sup.n -1], then [SIGMA] is an n -manifold (Proposition 7.3.7).
There is great freedom of choice for the simplicial complex L . As we shall see in Lemma 7.2.2, if L is the barycentric subdivision of any finite polyhedral cell complex, we can find a Coxeter system with nerve L . So, the topological type of L is completely arbitrary. This arbitrariness is the source of power for the using Coxeter groups to construct interesting examples in geometric and combinatorial group theory.
Coxeter Groups as a Source of Examples in Geometric and Combinatorial Group Theory
Here are some of the examples.
The Eilenberg-Ganea Problem asks if every group [pi] of cohomological dimension 2 has a two-dimensional model for its classifying space B[pi] (defined in 2.3). It is known that the minimum dimension of a model for B[pi] is either 2 or 3. Suppose L is a two-dimensional acyclic complex with [pi].sub.1 [not equal to] 1. Conjecturally, any torsion-free subgroup of finite index in W should be a counterexample to the Eilenberg-Ganea Problem (see Remark 8.5.7). Although the Eilenberg-Ganea Problem is still open, it is proved in that W is a counterexample to the appropriate version of it for groups with torsion. More precisely, the lowest possible dimension for any [bar.E]W is 3 (= dim [SIGMA]) while the algebraic version of this dimension is 2.
Suppose L is a triangulation of the real projective plane. If [GAMMA] [subset] W is a torsion-free subgroup of finite index, then its cohomological dimension over Z is 3 but over Q is 2 (see Section 8.5).
Suppose L is a triangulation of a homology ( n - 1)-sphere, n [greater than or equal to] 4, with [pi.sub.1 [not equal to] 1. It is shown in that a slight modification of [SIGMA] gives a contractible n -manifold not homeomorphic to [R.sup. n ]. This gave the first examples of closed apherical manifolds not covered by Euclidean space. Later, it was proved in that by choosing L to be an appropriate "generalized homology sphere," it is not necessary to modify [SIGMA]; it is already a CAT(0)-manifold not homeomorphic to Euclidean space. (Such examples are discussed in Chapter 10.)
The Reflection Group Trick
This a technique for converting finite aspherical CW complexes into closed aspherical manifolds. The main consequence of the trick is the following.
Theorem.(Theorem 11.1). Suppose [pi] is a group so that B[pi] is homotopy equivalent to a finite CW complex. Then there is a closed aspherical manifold M which retracts onto B[pi].
This trick yields a much larger class of groups than Coxeter groups. The group that acts on the universal cover of M is a semidirect product [??] [??] [pi], where [??] is an (infinitely generated) Coxeter group. In Chapter 11 this trick is used to produce a variety examples. These examples answer in the negative many of questions about aspherical manifolds raised in Wall's list of problems in. By using the above theorem, one can construct examples of closed aspherical manifolds M where [pi].sub.1 (a) is not residually finite, (b) contains infinitely divisible abelian subgroups, or (c) has unsolvable word problems. In 11.3, following, we use the reflection group trick to produce examples of closed aspherical topological manifolds not homotopy equivalent to closed smooth manifolds. In 11.4 we use the trick to show that if the Borel Conjecture (from surgery theory) holds for all groups [pi] which are fundamental groups of closed aspherical manifolds, then it must also hold for any [pi] with a finite classifying space. In 11.5 we combine a version of the reflection group trick with the examples of Bestvina and Brady in [24] to show that there are Poincar duality groups which are not finitely presented. (Hence, there are Poincarduality groups which do not arise as fundamental groups of closed aspherical manifolds.)
Description:
The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.
Review
"This book is one of those that grows with the reader: A graduate student can learn many properties, details and examples of Coxeter groups, while an expert can read about aspects of recent results in the theory of Coxeter groups and use the book as a guide to the literature. I strongly recommend this book to anybody who has any interest in geometric group theory. Anybody who reads (parts of) this book with an open mind will get a lot out of it." ---Ralf Gramlich, Mathematical Reviews
" The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory." ― L'Enseignement Mathematique
"[An] excellent introduction to other, important aspects of the study of geometric and topological approaches to group theory. Davis's exposition gives a delightful treatment of infinite Coxeter groups that illustrates their continued utility to the field." ---John Meier, Bulletin of the AMS
Review
"This is a comprehensive―nearly encyclopedic―survey of results concerning Coxeter groups. No other book covers the more recent important results, many of which are due to Michael Davis himself. This is an excellent, thoughtful, and well-written book, and it should have a wide readership among pure mathematicians in geometry, topology, representation theory, and group theory." ―Graham A. Niblo, University of Southampton
"Davis's book is a significant addition to the mathematics literature and it provides an important access point for geometric group theory. Although the book is a focused research monograph, it does such a nice job of presenting important material that it will also serve as a reference for quite some time. In fact, for years to come mathematicians will be writing 'terminology and notation follow Davis' in the introductions to papers on the geometry and topology of infinite Coxeter groups." ―John Meier, Lafayette College
From the Inside Flap
"This is a comprehensive--nearly encyclopedic--survey of results concerning Coxeter groups. No other book covers the more recent important results, many of which are due to Michael Davis himself. This is an excellent, thoughtful, and well-written book, and it should have a wide readership among pure mathematicians in geometry, topology, representation theory, and group theory." --Graham A. Niblo, University of Southampton
"Davis's book is a significant addition to the mathematics literature and it provides an important access point for geometric group theory. Although the book is a focused research monograph, it does such a nice job of presenting important material that it will also serve as a reference for quite some time. In fact, for years to come mathematicians will be writing 'terminology and notation follow Davis' in the introductions to papers on the geometry and topology of infinite Coxeter groups." --John Meier, Lafayette College
From the Back Cover
"This is a comprehensive--nearly encyclopedic--survey of results concerning Coxeter groups. No other book covers the more recent important results, many of which are due to Michael Davis himself. This is an excellent, thoughtful, and well-written book, and it should have a wide readership among pure mathematicians in geometry, topology, representation theory, and group theory." --Graham A. Niblo, University of Southampton
"Davis's book is a significant addition to the mathematics literature and it provides an important access point for geometric group theory. Although the book is a focused research monograph, it does such a nice job of presenting important material that it will also serve as a reference for quite some time. In fact, for years to come mathematicians will be writing 'terminology and notation follow Davis' in the introductions to papers on the geometry and topology of infinite Coxeter groups." --John Meier, Lafayette College
About the Author
Michael W. Davis is professor of mathematics at Ohio State University.
Excerpt. © Reprinted by permission. All rights reserved.
The Geometry and Topology of Coxeter Groups
By Michael W. Davis
Princeton University Press
Copyright © 2007 Princeton University Press
All right reserved.
ISBN: 978-0-691-13138-2
Chapter One
INTRODUCTION AND PREVIEW
1.1. INTRODUCTION
Geometric Reflection Groups
Finite groups generated by orthogonal linear reflections on [R.sup. n ] play a decisive role in
the classification of Lie groups and Lie algebras;
the theory of algebraic groups, as well as, the theories of spherical buildings and finite groups of Lie type;
the classification of regular polytopes (see or Appendix B).
Finite reflection groups also play important roles in many other areas of mathematics, e.g., in the theory of quadratic forms and in singularity theory. We note that a finite reflection group acts isometrically on the unit sphere [S.sup. n -1] of [R.sup. n ].
There is a similar theory of discrete groups of isometries generated by affine reflections on Euclidean space [E.sup. n ]. When the action of such a Euclidean reflection group has compact orbit space it is called cocompact . The classification of cocompact Euclidean reflection groups is important in Lie theory, in the theory of lattices in [R.sup. n ] and in E. Cartan's theory of symmetric spaces. The classification of these groups and of the finite (spherical) reflection groups can be found in Coxeter's 1934 paper. We give this classification in Table 6.1 of Section 6.9 and its proof in Appendix C.
There are also examples of discrete groups generated by reflections on the other simply connected space of constant curvature, hyperbolic n -space, [H.sup. n ]. (See [257, 291] as well as Chapter 6 for the theory of hyperbolic reflection groups.)
The other symmetric spaces do not admit such isometry groups. The reason is that the fixed set of a reflection should be a submanifold of codimension one (because it must separate the space) and the other (irreducible) symmetric spaces do not have codimension-one, totally geodesic subspaces. Hence, they do not admit isometric reflections. Thus, any truly "geometric" reflection group must split as a product of spherical, Euclidean, and hyperbolic ones.
The theory of these geometric reflection groups is the topic of Chapter 6. Suppose W is a reflection group acting on [X.sup. n ] = [S.sup. n ], [E.sup. n ], or [H.sup. n ]. Let K be the closure of a connected component of the complement of the union of "hyperplanes" which are fixed by some reflection in W . There are several common features to all three cases:
K is geodesically convex polytope in [X.sup. n ].
K is a "strict" fundamental domain in the sense that it intersects each orbit in exactly one point (so, [X.sup. n ]/ W [congruent to] = K ).
If S is the set of reflections across the codimension-one faces of K , then each reflection in W is conjugate to an element of S (and hence, S generates W ).
Abstract Reflection Groups
The theory of abstract reflection groups is due to Tits. What is the appropriate notion of an "abstract reflection group"? At first approximation, one might consider pairs (W, S) , where W is a group and S is any set of involutions which generates W . This is obviously too broad a notion. Nevertheless, it is a step in the right direction. In Chapter 3, we shall call such a pair a "pre-Coxeter system." There are essentially two completely different definitions for a pre-Coxeter system to be an abstract reflection group.
The first focuses on the crucial feature that the fixed point set of a reflection should separate the ambient space. One version is that the fixed point set of each element of S separates the Cayley graph of (W, S) (defined in Section 2.1). In 3.2 we call (W, S) a reflection system if it satisfies this condition. Essentially, this is equivalent to any one of several well-known combinatorial conditions, e.g., the Deletion Condition or the Exchange Condition. The second definition is that (W, S) has a presentation of a certain form. Following Tits, a pre-Coxeter system with such a presentation is a "Coxeter system" and W a "Coxeter group." Remarkably, these two definitions are equivalent. This was basically proved in. Another proof can be extracted from the first part of Bourbaki. It is also proved as the main result (Theorem 3.3.4) of Chapter 3. The equivalence of these two definitions is the principal mechanism driving the combinatorial theory of Coxeter groups.
The details of the second definition go as follows. For each pair (s, t) [member of] S x S , let [m.sub.st] denote the order of st . The matrix ( [m.sub.st] ) is the Coxeter matrix of (W, S) ; it is a symmetric S x S **matrix with entries in N [union] {[infinity]}, 1's on the diagonal, and each off-diagonal entry > 1. Let **
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(W, S) is a Coxeter system if <S|R > is a presentation for W . It turns out that, given any S x S matrix ( [m.sub.st] ) as above, the group W defined by the presentation <S|R > gives a Coxeter system (W, S) . (This is Corollary 6.12.6 of Chapter 6.)
Geometrization of Abstract Reflection Groups
Can every Coxeter system (W, S) be realized as a group of automorphisms of an appropriate geometric object? One answer was provided by Tits: for any (W, S) , there is a faithful linear representation W [right arrow] GL(N, R) , with N = Card(S), so that
Each element of S is represented by a linear reflection across a codimension-one face of a simplicial cone ITLITL. (N.B. A "linear reflection" means a linear involution with fixed subspace of codimension one; however, no inner product is assumed and the involution is not required to be orthogonal.)
If w [member of] W and w [not equal to] 1, then w (int(ITLITL)) [intersection] int(ITLITL) = (here int(ITLITL) denotes the interior of ITLITL).
WC , the union of W -translates of ITLITL, is a convex cone.
W acts properly on the interior I of WC .
Let [C.sup.f] := I [intersection] ITLITL. Then [C.sup.f] is the union of all (open) faces of ITLITL which have finite stabilizers (including the face int(ITLITL)). Moreover, [C.sup.f] is a strict fundamental domain for W on I .
Proofs of the above facts can be found in Appendix D. Tits' result was extended by Vinberg, who showed that for many Coxeter systems there are representations of W on [R.sup. N ], with N S)and ITLITL a polyhedral cone which is not simplicial. However, the poset of faces with finite stabilizers is exactly the same in both cases: it is the opposite poset to the poset of subsets of S which generate finite subgroups of W . (These are the "spherical subsets" of Definition 7.1.1 in Chapter 7.) The existence of Tits' geometric representation has several important consequences. Here are two:
Any Coxeter group W is virtually torsion-free.
I (the interior of the Tits cone) is a model for [E.bar]W , the "universal space for proper W -actions" (defined in 2.3).
Tits gave a second geometrization of (W, S) : its "Coxeter complex" [XI]. This is a certain simplicial complex with W -action. There is a simplex [sigma] [subset] [XI] with dim [sigma] = Card( S ) - 1 such that (a) [sigma] is a strict fundamental domain and (b) the elements of S act as "reflections" across the codimension-one faces of [sigma]. When W is finite, [XI] is homeomorphic to unit sphere [S.sup. n -1] in the canonical representation, triangulated by translates of a fundamental simplex. When (W, S) arises from an irreducible cocompact reflection group on [E.sup. n ], [XI] [congruent to] [E.sup. n ]. It turns out that [XI] is contractible whenever W is infinite.
The realization of (W, S) as a reflection group on the interior I of the Tits cone is satisfactory for several reasons; however, it lacks two advantages enjoyed by the geometric examples on spaces of constant curvature:
The W -action on I is not cocompact (i.e., the strict fundamental domain [C.sup.f] is not compact).
There is no natural metric on I that is preserved by W . (However, in [200] McMullen makes effective use of a "Hilbert metric" on I .)
In general, the Coxeter complex also has a serious defect-the isotropy subgroups of the W -action need not be finite (so the W -action need not be proper). One of the major purposes of this book is to present an alternative geometrization for (W, S) which remedies these difficulties. This alternative is the cell complex [SIGMA], discussed below and in greater detail in Chapters 7 and 12 (and many other places throughout the book).
The Cell Complex [SIGMA]
Given a Coxeter system (W, S) , in Chapter 7 we construct a cell complex [SIGMA] with the following properties:
The 0-skeleton of [SIGMA] is W .
The 1-skeleton of [SIGMA] is Cay (W, S) , the Cayley graph of 2.1.
The 2-skeleton of [SIGMA] is a Cayley 2-complex (defined in 2.2) associated to the presentation <S|R **>. **
[SIGMA] has one W -orbit of cells for each spherical subset T [subset] S . The dimension of a cell in this orbit is Card( T ). In particular, if W is finite, [SIGMA] is a convex polytope.
W acts properly on [SIGMA].
W acts cocompactly on [SIGMA] and there is a strict fundamental domain K .
[SIGMA] is a model for [E.bar]W . In particular, it is contractible.
If (W, S) is the Coxeter system underlying a cocompact geometric reflection group on [X.sup. n ] = [E.sup. n ] or [H.sup. n ], then [SIGMA] is W -equivariantly homeomorphic to [X.sup. n ] and K is isomorphic to the fundamental polytope.
Moreover, the cell structure on [SIGMA] is dual to the cellulation of [X.sup. n ] by translates of the fundamental polytope.
The elements of S act as "reflections" across the "mirrors" of K . (In the geometric case where K is a polytope, a mirror is a codimension-one face.)
[SIGMA] embeds in I and there is a W -equivariant deformation retraction from I onto [SIGMA]. So [SIGMA] is the "cocompact core" of I .
There is a piecewise Euclidean metric on [SIGMA] (in which each cell is identified with a convex Euclidean polytope) so that W acts via isometries. This metric is CAT(0) in the sense of Gromov. (This gives an alternative proof that [SIGMA] is a model for [bar.E]W .)
The last property is the topic of Chapter 12 and Appendix I. In the case of "right-angled" Coxeter groups, this CAT(0) property was established by Gromov. ("Right angled" means that [m.sub.st] = 2 or [infinity] whenever s [not equal to] t.) Shortly after the appearance of, Moussong proved in his Ph.D. thesis that [SIGMA] is CAT(0) for any Coxeter system. The complexes [SIGMA] gave one of the first large class of examples of "CAT(0)-polyhedra" and showed that Coxeter groups are examples of "CAT(0)-groups." This is the reason why Coxeter groups are important in geometric group theory. Moussong's result also allowed him to find a simple characterization of when Coxeter groups are word hyperbolic in the sense of (Theorem 12.6.1).
Since W acts simply transitively on the vertex set of [SIGMA], any two vertices have isomorphic neighborhoods. We can take such a neighborhood to be the cone on a certain simplicial complex L , called the "link" of the vertex. (See Appendix A.6.) We also call L the "nerve" of (W, S) . It has one simplex for each nonempty spherical subset T [subset] S . (The dimension of the simplex is Card( T ) - 1.)If L is homeomorphic to [ S.sup.n -1], then [SIGMA] is an n -manifold (Proposition 7.3.7).
There is great freedom of choice for the simplicial complex L . As we shall see in Lemma 7.2.2, if L is the barycentric subdivision of any finite polyhedral cell complex, we can find a Coxeter system with nerve L . So, the topological type of L is completely arbitrary. This arbitrariness is the source of power for the using Coxeter groups to construct interesting examples in geometric and combinatorial group theory.
Coxeter Groups as a Source of Examples in Geometric and Combinatorial Group Theory
Here are some of the examples.
The Eilenberg-Ganea Problem asks if every group [pi] of cohomological dimension 2 has a two-dimensional model for its classifying space B[pi] (defined in 2.3). It is known that the minimum dimension of a model for B[pi] is either 2 or 3. Suppose L is a two-dimensional acyclic complex with [pi].sub.1 [not equal to] 1. Conjecturally, any torsion-free subgroup of finite index in W should be a counterexample to the Eilenberg-Ganea Problem (see Remark 8.5.7). Although the Eilenberg-Ganea Problem is still open, it is proved in that W is a counterexample to the appropriate version of it for groups with torsion. More precisely, the lowest possible dimension for any [bar.E]W is 3 (= dim [SIGMA]) while the algebraic version of this dimension is 2.
Suppose L is a triangulation of the real projective plane. If [GAMMA] [subset] W is a torsion-free subgroup of finite index, then its cohomological dimension over Z is 3 but over Q is 2 (see Section 8.5).
Suppose L is a triangulation of a homology ( n - 1)-sphere, n [greater than or equal to] 4, with [pi.sub.1 [not equal to] 1. It is shown in that a slight modification of [SIGMA] gives a contractible n -manifold not homeomorphic to [R.sup. n ]. This gave the first examples of closed apherical manifolds not covered by Euclidean space. Later, it was proved in that by choosing L to be an appropriate "generalized homology sphere," it is not necessary to modify [SIGMA]; it is already a CAT(0)-manifold not homeomorphic to Euclidean space. (Such examples are discussed in Chapter 10.)
The Reflection Group Trick
This a technique for converting finite aspherical CW complexes into closed aspherical manifolds. The main consequence of the trick is the following.
Theorem. (Theorem 11.1). Suppose [pi] is a group so that B[pi] is homotopy equivalent to a finite CW complex. Then there is a closed aspherical manifold M which retracts onto B[pi].
This trick yields a much larger class of groups than Coxeter groups. The group that acts on the universal cover of M is a semidirect product [??] [??] [pi], where [??] is an (infinitely generated) Coxeter group. In Chapter 11 this trick is used to produce a variety examples. These examples answer in the negative many of questions about aspherical manifolds raised in Wall's list of problems in. By using the above theorem, one can construct examples of closed aspherical manifolds M where [pi].sub.1 (a) is not residually finite, (b) contains infinitely divisible abelian subgroups, or (c) has unsolvable word problems. In 11.3, following, we use the reflection group trick to produce examples of closed aspherical topological manifolds not homotopy equivalent to closed smooth manifolds. In 11.4 we use the trick to show that if the Borel Conjecture (from surgery theory) holds for all groups [pi] which are fundamental groups of closed aspherical manifolds, then it must also hold for any [pi] with a finite classifying space. In 11.5 we combine a version of the reflection group trick with the examples of Bestvina and Brady in [24] to show that there are Poincar duality groups which are not finitely presented. (Hence, there are Poincarduality groups which do not arise as fundamental groups of closed aspherical manifolds.)
(Continues...)
Excerpted from The Geometry and Topology of Coxeter Groups by Michael W. Davis Copyright © 2007 by Princeton University Press . Excerpted by permission.
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