BuildingsFor 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
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 |
207 | |
211 | |
212 | |
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Common terms and phrases
action adjacent affine apartment containing apartment system arbitrary associated assume automorphism axioms bijection BN-pair Bruhat Bruhat decomposition C₁ called canonical chamber complex chamber map Chapter codimension compact consisting convex Coxeter complex Coxeter diagram Coxeter group Coxeter matrix Coxeter system defined dimension double cosets equivalent Euclidean building Euclidean reflection group example Exercise fact finite reflection group fixes flag complex folding follows fundamental chamber geometry given group scheme group W half-spaces hence hyperplanes implies incidence geometry inequality inner product irreducible isometry isomorphism labelling Lemma linear algebraic group maximal minimal gallery monomial matrix non-empty non-zero Note parabolic subgroups pointwise poset proof Proposition prove reduced decomposition retraction root system simplex simplicial complex special cosets special subgroup spherical building stabilizer subcomplex subgroup of G subsector subset subspaces Suppose system of apartments theorem type-preserving V₁ valuation vector space vertex vertices walls Weyl group