Geometric And Topological Aspects Of Coxeter Groups And Buildings (zurich Lectures In Advanced Mathematics)

Coxeter groups are groups generated by reflections, and they appear throughout mathematics. Tits developed the general theory of Coxeter groups in order to develop the theory of buildings. Buildings have interrelated algebraic, combinatorial and geometric structures, and are powerful tools for understanding the groups which act on them. These notes focus on the geometry and topology of Coxeter groups and buildings, especially nonspherical cases. The emphasis is on geometric intuition, and there are many examples and illustrations. Part I describes Coxeter groups and their geometric realisations, particularly the Davis complex, and Part II gives a concise introduction to buildings. This book will be suitable for mathematics graduate students and researchers in geometric group theory, as well as algebra and combinatorics. The assumed background is basic group theory, including group actions, and basic algebraic topology, together with some knowledge of Riemannian geometry. Keywords: Coxeter groups, buildings, Davis complexes Coxeter groups......Page 14 Geometric reflection groups......Page 16 Definition of a Coxeter group......Page 28 Right-angled Coxeter groups......Page 30 Weyl groups......Page 31 Word metrics and Cayley graphs......Page 34 Cayley graphs of Coxeter systems......Page 36 Reflection systems......Page 38 Coxeter and reflection systems; deletion and exchange conditions......Page 40 Construction of the Tits representation......Page 48 Geometry when m_{ij}=infty......Page 51 Faithfulness of the Tits representation......Page 53 Discreteness and linearity......Page 55 Geometric realisations of finite and affine Coxeter groups......Page 56 Special subgroups......Page 58 Motivation for other geometric realisations......Page 59 Simplicial complexes......Page 60 The basic construction......Page 61 Properties of the basic construction......Page 65 Action of W on the basic construction......Page 67 Universal property of the basic construction......Page 69 The basic construction and geometric reflection groups......Page 70 Spherical special subgroups and the nerve......Page 74 The Davis complex as a basic construction......Page 77 Contractibility of the Davis complex......Page 83 The Davis complex as the geometric realisation of a poset......Page 85 The Davis complex as a CW complex......Page 87 The Davis complex is CAT(0)......Page 90 When is the Davis complex CAT(-1)?......Page 96 Cohomology of Coxeter groups and applications......Page 99 Buildings......Page 104 Buildings as unions of apartments......Page 106 First examples of buildings......Page 108 Extended example: The building for GL_3(q)......Page 112 Chamber systems and related notions......Page 118 Buildings as chamber systems......Page 120 Equivalence of definitions......Page 121 Comparing the definitions......Page 124 Right-angled buildings......Page 125 Definition of retractions......Page 128 Examples of retractions......Page 129 Applications of retractions......Page 130 BN-pairs and the Bruhat decomposition......Page 134 Strongly transitive actions......Page 136 The building associated to a BN-pair......Page 137 Parabolic subgroups......Page 139 Spherical, affine and Kac–Moody BN-pairs......Page 140 Links in buildings......Page 150 Constructions of exotic buildings......Page 151 References......Page 154 Index......Page 158 Coxeter groups are groups generated by reflections. They appear throughout mathematics. Tits developed the general theory of Coxeter groups in order to develop the theory of buildings. Buildings have interrelated algebraic, combinatorial and geometric structures and are powerful tools for understanding the groups which act on them. These notes focus on the geometry and topology of Coxeter groups and buildings, especially nonspherical cases. The emphasis is on geometric intuition, and there are many examples and illustrations. Part I describes Coxeter groups and their geometric realizations, particularly the Davis complex, and Part II gives a concise introduction to buildings. This book will be suitable for graduate students and researchers in geometric group theory, as well as algebra and combinatorics. The assumed background is basic group theory, including group actions, and basic algebraic topology, together with some knowledge of Riemannian geometry.