THE GEOMETRY AND TOPOLOGY OF COXETER GROUPS

By EMILY LEWIS

CONTENTS CHAPTER 1 INTRODUCTION AND PREVIEW INTRODUCTION A PREVIEW OF THE RIGHT-ANGLED CASE CHAPTER 2 SOME BASIC NOTIONS IN GEOMETRIC GROUP THEORY CAYLEY GRAPHS AND WORD METRICS CAYLEY 2-COMPLEXES BACKGROUND ON ASPHERICAL SPACES CHAPTER 3 COXETER GROUPS DIHEDRAL GROUPS RE?ECTION SYSTEMS COXETER SYSTEMS THE WORD PROBLEM COXETER DIAGRAMS CHAPTER 4 MORE COMBINATORIAL THEORY OF COXETER GROUPS SPECIAL SUBGROUPS IN COXETER GROUPS RE?ECTIONS THE SHORTEST ELEMENT IN A SPECIAL COSET ANOTHER CHARACTERIZATION OF COXETER GROUPS CONVEX SUBSETS OF W THE ELEMENT OF LONGEST LENGTH THE LETTERS WITH WHICH A REDUCED EXPRESSION CAN END A LEMMA OF TITS SUBGROUPS GENERATED BY RE?ECTIONS NORMALIZERS OF SPECIAL SUBGROUPS CHAPTER 5 THE BASIC CONSTRUCTION THE SPACE U THE CASE OF A PRE-COXETER SYSTEM SECTORS IN U CHAPTER 6 GEOMETRIC REFLECTION GROUPS LINEAR RE?ECTIONS SPACES OF CONSTANT CURVATURE POLYTOPES WITH NONOBTUSE DIHEDRAL ANGLES THE DEVELOPING MAP POLYGON GROUPS FINITE LINEAR GROUPS GENERATED BY RE?ECTIONS EXAMPLES OF FINITE RE?ECTION GROUPS GEOMETRIC SIMPLICES: THE GRAM MATRIX AND THE COSINE MATRIX SIMPLICIAL COXETER GROUPS: LANN´ER’S THEOREM THREE-DIMENSIONAL HYPERBOLIC RE?ECTION GROUPS: ANDREEV’S THEOREM HIGHER-DIMENSIONAL HYPERBOLIC RE?ECTION GROUPS: VINBERG’S THEOREM THE CANONICAL REPRESENTATION CHAPTER 7 THE COMPLEX THE NERVE OF A COXETER SYSTEM GEOMETRIC REALIZATIONS A CELL STRUCTURE ON EXAMPLES FIXED POSETS AND FIXED SUBSPACES CHAPTER 8 THE ALGEBRAIC TOPOLOGY OF U AND OF THE HOMOLOGY OF U ACYCLICITY CONDITIONS COHOMOLOGY WITH COMPACT SUPPORTS THE CASE WHERE X IS A GENERAL SPACE COHOMOLOGY WITH GROUP RING COEF?CIENTS BACKGROUND ON THE ENDS OF A GROUP THE ENDS OF W SPLITTINGS OF COXETER GROUPS COHOMOLOGY OF NORMALIZERS OF SPHERICAL SPECIAL SUBGROUPS CHAPTER 9 THE FUNDAMENTAL GROUP AND THE FUNDAMENTAL GROUP AT INFINITY THE FUNDAMENTAL GROUP OF U WHAT IS SIMPLY CONNECTED AT IN?NITY? CHAPTER 10 ACTIONS ON MANIFOLDS RE?ECTION GROUPS ON MANIFOLDS THE TANGENT BUNDLE BACKGROUND ON CONTRACTIBLE MANIFOLDS BACKGROUND ON HOMOLOGY MANIFOLDS ASPHERICAL MANIFOLDS NOT COVERED BY EUCLIDEAN SPACE WHEN IS A MANIFOLD? RE?ECTION GROUPS ON HOMOLOGY MANIFOLDS GENERALIZED HOMOLOGY SPHERES AND POLYTOPES VIRTUAL POINCAR´E DUALITY GROUPS CHAPTER 11 THE REFLECTION GROUP TRICK THE FIRST VERSION OF THE TRICK EXAMPLES OF FUNDAMENTAL GROUPS OF CLOSED ASPHERICAL MANIFOLDS NONSMOOTHABLE ASPHERICAL MANIFOLDS THE BOREL CONJECTURE AND THE PDN-GROUP CONJECTURE THE SECOND VERSION OF THE TRICK THE BESTVINA-BRADY EXAMPLES THE EQUIVARIANT RE?ECTION GROUP TRICK CHAPTER 12 IS CAT(0): THEOREMS OF GROMOV AND MOUSSONG A PIECEWISE EUCLIDEAN CELL STRUCTURE ON THE RIGHT-ANGLED CASE THE GENERAL CASE THE VISUAL BOUNDARY OF BACKGROUND ON WORD HYPERBOLIC GROUPS WHEN IS CAT(?1)? FREE ABELIAN SUBGROUPS OF COXETER GROUPS RELATIVE HYPERBOLIZATION CHAPTER 13 RIGIDITY DE?NITIONS, EXAMPLES, COUNTEREXAMPLES SPHERICAL PARABOLIC SUBGROUPS AND THEIR FIXED SUBSPACES COXETER GROUPS OF TYPE PM STRONG RIGIDITY FOR GROUPS OF TYPE PM CHAPTER 14 FREE QUOTIENTS AND SURFACE SUBGROUPS LARGENESS SURFACE SUBGROUPS CHAPTER 15 ANOTHER LOOK AT (CO)HOMOLOGY COHOMOLOGY WITH CONSTANT COEF?CIENTS DECOMPOSITIONS OF COEF?CIENT SYSTEMS THE W-MODULE STRUCTURE ON (CO)HOMOLOGY THE CASE WHERE W IS ?NITE CHAPTER 16 THE EULER CHARACTERISTIC BACKGROUND ON EULER CHARACTERISTICS THE EULER CHARACTERISTIC CONJECTURE THE FLAG COMPLEX CONJECTURE CHAPTER 17 GROWTH SERIES RATIONALITY OF THE GROWTH SERIES EXPONENTIAL VERSUS POLYNOMIAL GROWTH RECIPROCITY RELATIONSHIP WITH THE H-POLYNOMIAL CHAPTER 18 BUILDINGS THE COMBINATORIAL THEORY OF BUILDINGS THE GEOMETRIC REALIZATION OF A BUILDING BUILDINGS ARE CAT(0) EULER-POINCAR´E MEASURE CHAPTER 19 HECKE–VON NEUMANN ALGEBRAS HECKE ALGEBRAS HECKE–VON NEUMANN ALGEBRAS CHAPTER 20 WEIGHTED L2-(CO)HOMOLOGY WEIGHTED L2-(CO)HOMOLOGY WEIGHTED L2-BETTI NUMBERS AND EULER CHARACTERISTICS CONCENTRATION OF (CO)HOMOLOGY IN DIMENSION WEIGHTED POINCAR´E DUALITY A WEIGHTED VERSION OF THE SINGER CONJECTURE DECOMPOSITION THEOREMS DECOUPLING COHOMOLOGY L2-COHOMOLOGY OF BUILDINGS