Buildings

Front Cover
Springer Science & Business Media, Jun 29, 2013 - Mathematics - 215 pages
For years I have heard about buildings and their applications to group theory. I finally decided to try to learn something about the subject by teaching a graduate course on it at Cornell University in Spring 1987. This book is based on the not es from that course. The course started from scratch and proceeded at a leisurely pace. The book therefore does not get very far. Indeed, the definition of the term "building" doesn't even appear until Chapter IV. My hope, however, is that the book gets far enough to enable the reader to tadle the literat ure on buildings, some of which can seem very forbidding. Most of the results in this book are due to J. Tits, who originated the the ory of buildings. The main exceptions are Chapter I (which presents some classical material), Chapter VI (which prcsents joint work of F. Bruhat and Tits), and Chapter VII (which surveys some applications, due to var ious people). It has been a pleasure studying Tits's work; I only hope my exposition does it justice.
 

Contents

Finite Reflection Groups
1
Coxeter Complexes
58
Buildings
76
The Axioms for a Thick Building
97
The Building Associated to a BNPair
112
The General Linear Group
118
The Special Linear Group Over a Field With
127
Euclidean Buildings
139
A Metric Characterization of the Apartments
165
Construction of Apartments
169
The Spherical Building at Infinity
174
Applications to Group Cohomology
183
SArithmetic Groups
189
Cohomological Dimension of Linear Groups
194
SArithmetic Groups Over Function Fields
195
Appendix Linear Algebraic Groups
198

Euclidean Coxeter Complexes
149
Euclidean Buildings as Metric Spaces
151
The BruhatTits FixedPoint Theorem
157
Bounded Subgroups
159
Bounded Subsets of Apartments
163
Suggestions for Further Reading
206
References
207
Notation Index
211
Subject Index
212
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