Introduces uniform constructions of most of the known compactifications of symmetric and locally symmetric spaces, with emphasis on their geometric and topological structures Relatively self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to analysis, number theory, algebraic geometry and algebraic topology
Author(s): Armand Borel, Lizhen Ji
Edition: 1
Year: 2005
Language: English
Pages: 496
Cover......Page 1
Compactifications of Symmetric and Locally Symmetric Spaces......Page 4
Copyright - ISBN: 0817632476......Page 5
Contents......Page 8
Preface......Page 12
0 Introduction......Page 18
0.1 History of compactifications......Page 19
0.2 New points of view in this book......Page 31
0.3 Organization and outline of the book......Page 33
0.4 Topics related to the book but not covered and classification of references......Page 34
Part I: Compactifications of Riemannian Symmetric Spaces......Page 40
1 Review of Classical Compactifications of Symmetric Spaces......Page 44
I.1 Real parabolic subgroups......Page 45
I.2 Geodesic compactification and Tits building......Page 56
I.3 Karpelevič compactification......Page 67
I.4 Satake compactifications......Page 72
I.5 Baily-Borel compactification......Page 94
I.6 Furstenberg compactifications......Page 108
I.7 Martin compactifications......Page 114
2 Uniform Construction of Compactifications of Symmetric Spaces......Page 124
I.8 Formulation of the uniform construction......Page 125
I.9 Siegel sets and generalizations......Page 131
I.10 Uniform construction of the maximal Satake compactification......Page 140
I.11 Uniform construction of nonmaximal Satake compactifications......Page 146
I.12 Uniform construction of the geodesic compactification......Page 156
I.13 Uniform construction of the Martin compactification......Page 161
I.14 Uniform construction of the Karpelevič compactification......Page 166
I.15 Real Borel-Serre partial compactification......Page 176
I.16 Relations between the compactifications......Page 182
I.17 More constructions of the maximal Satake compactification......Page 185
I.18 Compactifications as a topological ball......Page 191
I.19 Dual-cell compactification and maximal Satake compactification as a manifold with corners......Page 199
Part II: Smooth Compactifications of Semisimple Symmetric Spaces......Page 216
4 Smooth Compactifications of Riemannian Symmetric Spaces G/K......Page 220
II.1 Gluing of manifolds with corners......Page 221
II.2 The Oshima compactification of G/K......Page 227
II.3 Generalities on semisimple symmetric spaces......Page 232
II.4 Some real forms H[sub(ε)] of H......Page 234
II.5 Galois Cohomology......Page 238
II.6 Orbits of G in (G/H)(R)......Page 241
II.7 Examples......Page 247
II.8 Generalities on semisimple symmetric spaces......Page 250
II.9 The DeConcini-Procesi wonderful compactification of G/H......Page 253
II.10 Real points of G/H......Page 256
II.11 A characterization of the involutions σ[sub(ε)]......Page 261
II.12 Preliminaries on semisimple symmetric spaces......Page 266
II.13 The Oshima-Sekiguchi compactification of G/K......Page 270
II.14 Comparison with \overline{G/H}^W\mathbb{R}......Page 274
Part III: Compactifications of Locally Symmetric Spaces......Page 280
9 Classical Compactifications of Locally Symmetric Spaces......Page 284
III.1 Rational parabolic subgroups......Page 285
III.2 Arithmetic subgroups and reduction theories......Page 294
III.3 Satake compactifications of locally symmetric spaces......Page 303
III.4 Baily-Borel compactification......Page 310
III.5 Borel-Serre compactification......Page 318
III.6 Reductive Borel-Serre compactification......Page 326
III.7 Toroidal compactifications......Page 331
10 Uniform Construction of Compactifications of Locally Symmetric Spaces......Page 340
III.8 Formulation of the uniform construction......Page 341
III.9 Uniform construction of the Borel-Serre compactification......Page 343
III.10 Uniform construction of the reductive Borel-Serre compactification......Page 355
III.11 Uniform construction of the maximal Satake compactification......Page 362
III.12 Tits compactification......Page 367
III.13 Borel-Serre compactification of homogeneous spaces Γ\G......Page 372
III.14 Reductive Borel-Serre compactification of homogeneous spaces Γ\G......Page 376
11 Properties of Compactifications of Locally Symmetric Spaces......Page 382
III.15 Relations between the compactifications......Page 383
III.16 Self-gluing of Borel-Serre compactification into Borel-Serre-Oshima compactification......Page 387
12 Subgroup Compactifications of Γ\G......Page 392
III.17 Maximal discrete subgroups and space of subgroups......Page 393
III.18 Subgroup compactification of Γ\G and Γ\X......Page 399
III.19 Spaces of flags in \mathbb{R}[sup(n)], flag lattices, and compactifications of SL(n, \mathbb{Z})\SL(n,\mathbb{R})......Page 403
13 Metric Properties of Compactifications of Locally Symmetric Spaces Γ\X......Page 416
III.20 Eventually distance-minimizing geodesics and geodesic compactification of Γ\X......Page 417
III.21 Rough geometry of Γ\X and Siegel conjecture on metrics on Siegel sets......Page 422
III.22 Hyperbolic compactifications and extension of holomorphic maps from the punctured disk to Hermitian locally symmetric spaces......Page 426
III.23 Continuous spectrum, boundaries of compactifications, and scattering geodesics of Γ\X......Page 433
References......Page 440
N......Page 468
X......Page 469
B......Page 476
C......Page 478
D......Page 481
F......Page 482
G......Page 483
H......Page 484
L......Page 485
M......Page 486
P......Page 488
R......Page 490
S......Page 492
T......Page 495
Z......Page 496
