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
Author(s): Anne Thomas
Series: Zurich Lectures in Advanced Mathematics
Publisher: European Mathematical Society
Year: 2018
Language: English
Pages: 161
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
