HYPERBOLIC MANIFOLDS AND DISCRETE GROUPS
CONTENTS CHAPTER 1 GEOMETRY AND TOPOLOGY 1 1 1 SET THEORETIC PRELIMINARIES 1 1 1 1 GENERAL NOTATION 1 1 1 2 GROWTH RATES OF FUNCTIONS 2 1 1 3 JENSEN S INEQUALITY 3 1 2 MEASURE AND INTEGRAL 3 1 2 1 MEASURES 3 1 2 2 FINITELY ADDITIVE INTEGRALS 5 1 3 TOPOLOGICAL SPACES LEBESGUE COVERING DIMENSION 7 1 4 EXHAUSTIONS OF LOCALLY COMPACT SPACES 10 1 5 DIRECT AND INVERSE LIMITS 11 1 6 GRAPHS 13 1 7 COMPLEXES AND HOMOLOGY 17 1 7 1 SIMPLICIAL COMPLEXES 17 1 7 2 CELL COMPLEXES 19 CHAPTER 2 METRIC SPACES 23 2 1 GENERAL METRIC SPACES 23 2 2 LENGTH METRIC SPACES 25 2 3 GRAPHS AS LENGTH SPACES 27 2 4 HAUSDOR AND GROMOV HAUSDOR DISTANCES NETS 28 2 5 LIPSCHITZ MAPS AND BANACH MAZUR DISTANCE 30 2 5 1 LIPSCHITZ AND LOCALLY LIPSCHITZ MAPS 30 2 5 2 BI LIPSCHITZ MAPS THE BANACH MAZUR DISTANCE 33 2 6 HAUSDOR DIMENSION 34 2 7 NORMS AND VALUATIONS 35 2 8 NORMS ON ELD EXTENSIONS ADELES 39 2 9 METRICS ON A NE AND PROJECTIVE SPACES 43 2 10 QUASIPROJECTIVE TRANSFORMATIONS PROXIMAL TRANSFORMATIONS 48 2 11 KERNELS AND DISTANCE FUNCTIONS 51 CHAPTER 3 DI ERENTIAL GEOMETRY 59 3 1 SMOOTH MANIFOLDS 59 3 2 SMOOTH PARTITION OF UNITY 61 3 3 RIEMANNIAN METRICS 61 3 4 RIEMANNIAN VOLUME 64 3 5 VOLUME GROWTH AND ISOPERIMETRIC FUNCTIONS CHEEGER CONSTANT 68 3 6 CURVATURE 71 3 7 RIEMANNIAN MANIFOLDS OF BOUNDED GEOMETRY 73 3 8 METRIC SIMPLICIAL COMPLEXES OF BOUNDED GEOMETRY AND SYSTOLIC INEQUALITIES 74 3 9 HARMONIC FUNCTIONS 79 3 10 SPECTRAL INTERPRETATION OF THE CHEEGER CONSTANT 82 3 11 COMPARISON GEOMETRY 82 3 11 1 ALEXANDROV CURVATURE AND CAT SPACES 82 3 11 2 CARTAN S FIXED POINT THEOREM 86 3 11 3 IDEAL BOUNDARY HOROBALLS AND HOROSPHERES 88 CHAPTER 4 HYPERBOLIC SPACE 91 4 1 MOEBIUS TRANSFORMATIONS 91 4 2 REAL HYPERBOLIC SPACE 94 4 3 CLASSI CATION OF ISOMETRIES 99 4 4 HYPERBOLIC TRIGONOMETRY 102 4 5 TRIANGLES AND CURVATURE OF HN 105 4 6 DISTANCE FUNCTION ON HN 108 4 7 HYPERBOLIC BALLS AND SPHERES 110 4 8 HOROBALLS AND HOROSPHERES IN HN 110 4 9 HN AS A SYMMETRIC SPACE 112 4 10 INSCRIBED RADIUS AND THINNESS OF HYPERBOLIC TRIANGLES 116 4 11 EXISTENCE UNIQUENESS THEOREM FOR TRIANGLES 118 CHAPTER 5 GROUPS AND THEIR ACTIONS 119 5 1 SUBGROUPS 120 5 2 VIRTUAL ISOMORPHISMS OF GROUPS AND COMMENSURATORS 122 5 3 COMMUTATORS AND THE COMMUTATOR SUBGROUP 124 5 4 SEMIDIRECT PRODUCTS AND SHORT EXACT SEQUENCES 126 5 5 DIRECT SUMS AND WREATH PRODUCTS 128 5 6 GEOMETRY OF GROUP ACTIONS 129 5 6 1 GROUP ACTIONS 129 5 6 2 LINEAR ACTIONS 133 5 6 3 LIE GROUPS 134 5 6 4 HAAR MEASURE AND LATTICES 137 5 6 5 GEOMETRIC ACTIONS 140 5 7 ZARISKI TOPOLOGY AND ALGEBRAIC GROUPS 140 5 8 GROUP ACTIONS ON COMPLEXES 147 5 8 1 G COMPLEXES 147 5 8 2 BOREL AND HAE IGER CONSTRUCTIONS 148 5 8 3 GROUPS OF NITE TYPE 159 5 9 COHOMOLOGY 160 5 9 1 GROUP RINGS AND MODULES 160 5 9 2 GROUP COHOMOLOGY 161 5 9 3 BOUNDED COHOMOLOGY OF GROUPS 165 5 9 4 RING DERIVATIONS 166 5 9 5 DERIVATIONS AND SPLIT EXTENSIONS 168 5 9 6 CENTRAL COEXTENSIONS AND SECOND COHOMOLOGY 171 CHAPTER 6 MEDIAN SPACES AND SPACES WITH MEASURED WALLS 175 6 1 MEDIAN SPACES 175 6 1 1 A REVIEW OF MEDIAN ALGEBRAS 176 6 1 2 CONVEXITY 177 6 1 3 EXAMPLES OF MEDIAN METRIC SPACES 178 6 1 4 CONVEXITY AND GATE PROPERTY IN MEDIAN SPACES 180 6 1 5 RECTANGLES AND PARALLEL PAIRS 182 6 1 6 APPROXIMATE GEODESICS AND MEDIANS COMPLETIONS OF MEDIAN SPACES 185 6 2 SPACES WITH MEASURED WALLS 186 6 2 1 DE NITION AND BASIC PROPERTIES 186 6 2 2 RELATIONSHIP BETWEEN MEDIAN SPACES AND SPACES WITH MEASURED WALLS 189 6 2 3 EMBEDDING A SPACE WITH MEASURED WALLS IN A MEDIAN SPACE 190 6 2 4 MEDIAN SPACES HAVE MEASURED WALLS 192 CHAPTER 7 FINITELY GENERATED AND NITELY PRESENTED GROUPS 199 7 1 FINITELY GENERATED GROUPS 199 7 2 FREE GROUPS 203 7 3 PRESENTATIONS OF GROUPS 206 7 4 THE RANK OF A FREE GROUP DETERMINES THE GROUP SUBGROUPS 212 7 5 FREE CONSTRUCTIONS AMALGAMS OF GROUPS AND GRAPHS OF GROUPS 213 7 5 1 AMALGAMS 213 7 5 2 GRAPHS OF GROUPS 214 7 5 3 CONVERTING GRAPHS OF GROUPS INTO AMALGAMS 216 7 5 4 TOPOLOGICAL INTERPRETATION OF GRAPHS OF GROUPS 216 7 5 5 CONSTRUCTING NITE INDEX SUBGROUPS 217 7 5 6 GRAPHS OF GROUPS AND GROUP ACTIONS ON TREES 219 7 6 PING PONG LEMMA EXAMPLES OF FREE GROUPS 222 7 7 FREE SUBGROUPS IN SU 2 226 7 8 PING PONG ON PROJECTIVE SPACES 226 7 9 CAYLEY GRAPHS 227 7 10 VOLUMES OF MAPS OF CELL COMPLEXES AND VAN KAMPEN DIAGRAMS 235 7 10 1 SIMPLICIAL CELLULAR AND COMBINATORIAL VOLUMES OF MAPS 235 7 10 2 TOPOLOGICAL INTERPRETATION OF NITE PRESENTABILITY 236 7 10 3 PRESENTATIONS OF CENTRAL COEXTENSIONS 236 7 10 4 DEHN FUNCTION AND VAN KAMPEN DIAGRAMS 238 7 11 RESIDUAL NITENESS 244 7 12 HOP AN AND CO HOP AN PROPERTIES 247 7 13 ALGORITHMIC PROBLEMS IN THE COMBINATORIAL GROUP THEORY 248 CHAPTER 8 COARSE GEOMETRY 251 8 1 QUASIISOMETRY 251 8 2 GROUP THEORETIC EXAMPLES OF QUASIISOMETRIES 261 8 3 A METRIC VERSION OF THE MILNOR SCHWARZ THEOREM 267 8 4 TOPOLOGICAL COUPLING 269 8 5 QUASIACTIONS 271 8 6 QUASIISOMETRIC RIGIDITY PROBLEMS 274 8 7 THE GROWTH FUNCTION 275 8 8 CODIMENSION ONE ISOPERIMETRIC INEQUALITIES 281 8 9 DISTORTION OF A SUBGROUP IN A GROUP 283 CHAPTER 9 COARSE TOPOLOGY 287 9 1 ENDS 287 9 1 1 THE NUMBER OF ENDS 287 9 1 2 THE SPACE OF ENDS 290 9 1 3 ENDS OF GROUPS 295 9 2 RIPS COMPLEXES AND COARSE HOMOTOPY THEORY 297 9 2 1 RIPS COMPLEXES 297 9 2 2 DIRECT SYSTEM OF RIPS COMPLEXES AND COARSE HOMOTOPY 299 9 3 METRIC CELL COMPLEXES 300 9 4 CONNECTIVITY AND COARSE CONNECTIVITY 306 9 5 RETRACTIONS 312 9 6 POINCAR DUALITY AND COARSE SEPARATION 314 9 7 METRIC LLING FUNCTIONS 317 9 7 1 COARSE ISOPERIMETRIC FUNCTIONS AND COARSE LLING RADIUS 318 9 7 2 QUASIISOMETRIC INVARIANCE OF COARSE LLING FUNCTIONS 320 9 7 3 HIGHER DEHN FUNCTIONS 325 9 7 4 COARSE BESIKOVITCH INEQUALITY 330 CHAPTER 10 ULTRALIMITS OF METRIC SPACES 333 10 1 THE AXIOM OF CHOICE AND ITS WEAKER VERSIONS 333 10 2 ULTRA LTERS AND THE STONE ECH COMPACTI CATION 339 10 3 ELEMENTS OF NON STANDARD ALGEBRA 340 10 4 ULTRALIMITS OF FAMILIES OF METRIC SPACES 344 10 5 COMPLETENESS OF ULTRALIMITS AND INCOMPLETENESS OF ULTRA LTERS 348 10 6 ASYMPTOTIC CONES OF METRIC SPACES 352 10 7 ULTRALIMITS OF ASYMPTOTIC CONES ARE ASYMPTOTIC CONES 356 10 8 ASYMPTOTIC CONES AND QUASIISOMETRIES 358 10 9 ASSOUAD TYPE THEOREMS 360 CHAPTER 11 GROMOV HYPERBOLIC SPACES AND GROUPS 363 11 1 HYPERBOLICITY ACCORDING TO RIPS 363 11 2 GEOMETRY AND TOPOLOGY OF REAL TREES 367 11 3 GROMOV HYPERBOLICITY 368 11 4 ULTRALIMITS AND STABILITY OF GEODESICS IN RIPS HYPERBOLIC SPACES 372 11 5 LOCAL GEODESICS IN HYPERBOLIC SPACES 376 11 6 QUASICONVEXITY IN HYPERBOLIC SPACES 379 11 7 NEAREST POINT PROJECTIONS 381 11 8 GEOMETRY OF TRIANGLES IN RIPS HYPERBOLIC SPACES 382 11 9 DIVERGENCE OF GEODESICS IN HYPERBOLIC METRIC SPACES 385 11 10 MORSE LEMMA REVISITED 387 11 11 IDEAL BOUNDARIES 390 11 12 GROMOV BORDI CATION OF GROMOV HYPERBOLIC SPACES 398 11 13 BOUNDARY EXTENSION OF QUASIISOMETRIES OF HYPERBOLIC SPACES 402 11 13 1 EXTENDED MORSE LEMMA 402 11 13 2 THE EXTENSION THEOREM 404 11 13 3 BOUNDARY EXTENSION AND QUASIACTIONS 406 11 13 4 CONICAL LIMIT POINTS OF QUASIACTIONS 407 11 14 HYPERBOLIC GROUPS 407 11 15 IDEAL BOUNDARIES OF HYPERBOLIC GROUPS 410 11 16 LINEAR ISOPERIMETRIC INEQUALITY AND DEHN ALGORITHM FOR HYPERBOLIC GROUPS 414 11 17 THE SMALL CANCELLATION THEORY 417 11 18 THE RIPS CONSTRUCTION 418 11 19 CENTRAL COEXTENSIONS OF HYPERBOLIC GROUPS AND QUASIISOMETRIES 419 11 20 CHARACTERIZATION OF HYPERBOLICITY USING ASYMPTOTIC CONES 423 11 21 SIZE OF LOOPS 429 11 21 1 THE MINSIZE 429 11 21 2 THE CONSTRICTION 430 11 22 FILLING INVARIANTS OF HYPERBOLIC SPACES 432 11 22 1 FILLING AREA 433 11 22 2 FILLING RADIUS 434 11 22 3 ORDERS OF DEHN FUNCTIONS OF NON HYPERBOLIC GROUPS AND HIGHER DEHN FUNCTIONS 437 11 23 ASYMPTOTIC CONES ACTIONS ON TREES AND ISOMETRIC ACTIONS ON HYPERBOLIC SPACES 438 11 24 SUMMARY OF EQUIVALENT DE NITIONS OF HYPERBOLICITY 441 11 25 FURTHER PROPERTIES OF HYPERBOLIC GROUPS 442 11 26 RELATIVELY HYPERBOLIC SPACES AND GROUPS 445 CHAPTER 12 LATTICES IN LIE GROUPS 449 12 1 SEMISIMPLE LIE GROUPS AND THEIR SYMMETRIC SPACES 449 12 2 LATTICES 451 12 3 EXAMPLES OF LATTICES 452 12 4 RIGIDITY AND SUPERRIGIDITY 454 12 5 COMMENSURATORS OF LATTICES 456 12 6 LATTICES IN PO N 1 456 12 6 1 ZARISKI DENSITY 456 12 6 2 PARABOLIC ELEMENTS AND NON COMPACTNESS 458 12 6 3 THICK THIN DECOMPOSITION 460 12 7 CENTRAL COEXTENSIONS 462 CHAPTER 13 SOLVABLE GROUPS 465 13 1 FREE ABELIAN GROUPS 465 13 2 CLASSI CATION OF NITELY GENERATED ABELIAN GROUPS 468 13 3 AUTOMORPHISMS OF ZN 471 13 4 NILPOTENT GROUPS 474 13 5 POLYCYCLIC GROUPS 484 13 6 SOLVABLE GROUPS DE NITION AND BASIC PROPERTIES 489 13 7 FREE SOLVABLE GROUPS AND THE MAGNUS EMBEDDING 491 13 8 SOLVABLE VERSUS POLYCYCLIC 493 CHAPTER 14 GEOMETRIC ASPECTS OF SOLVABLE GROUPS 497 14 1 WOLF S THEOREM FOR SEMIDIRECT PRODUCTS ZN Z 497 14 1 1 GEOMETRY OF H3 Z 499 14 1 2 DISTORTION OF SUBGROUPS OF SOLVABLE GROUPS 503