Geometric Group Theory

Cover......Page 1
Title page......Page 2
Contents......Page 6
Preface......Page 14
1.1.1. General notation......Page 22
1.1.2. Growth rates of functions......Page 23
1.2.1. Measures......Page 24
1.2.2. Finitely additive integrals......Page 26
1.3. Topological spaces. Lebesgue covering dimension......Page 28
1.4. Exhaustions of locally compact spaces......Page 31
1.5. Direct and inverse limits......Page 32
1.6. Graphs......Page 34
1.7.1. Simplicial complexes......Page 38
1.7.2. Cell complexes......Page 40
2.1. General metric spaces......Page 44
2.2. Length metric spaces......Page 46
2.3. Graphs as length spaces......Page 48
2.4. Hausdorff and Gromov–Hausdorff distances. Nets......Page 49
2.5.1. Lipschitz and locally Lipschitz maps......Page 51
2.5.2. Bi-Lipschitz maps. The Banach–Mazur distance......Page 54
2.6. Hausdorff dimension......Page 55
2.7. Norms and valuations......Page 56
2.8. Norms on field extensions. Adeles......Page 60
2.9. Metrics on affine and projective spaces......Page 64
2.10. Quasiprojective transformations. Proximal transformations......Page 69
2.11. Kernels and distance functions......Page 72
3.1. Smooth manifolds......Page 80
3.3. Riemannian metrics......Page 82
3.4. Riemannian volume......Page 85
3.5. Volume growth and isoperimetric functions. Cheeger constant......Page 89
3.6. Curvature......Page 92
3.7. Riemannian manifolds of bounded geometry......Page 94
3.8. Metric simplicial complexes of bounded geometry and systolic inequalities......Page 95
3.9. Harmonic functions......Page 100
3.11.1. Alexandrov curvature and ������(��) spaces......Page 103
3.11.2. Cartan’s Fixed-Point Theorem......Page 107
3.11.3. Ideal boundary, horoballs and horospheres......Page 109
4.1. Moebius transformations......Page 112
4.2. Real-hyperbolic space......Page 115
4.3. Classification of isometries......Page 120
4.4. Hyperbolic trigonometry......Page 123
4.5. Triangles and curvature of ℊⁿ......Page 126
4.6. Distance function on ℊⁿ......Page 129
4.8. Horoballs and horospheres in \bHⁿ......Page 131
4.9. \bHⁿ as a symmetric space......Page 133
4.10. Inscribed radius and thinness of hyperbolic triangles......Page 137
4.11. Existence-uniqueness theorem for triangles......Page 139
Chapter 5. Groups and their actions......Page 140
5.1. Subgroups......Page 141
5.2. Virtual isomorphisms of groups and commensurators......Page 143
5.3. Commutators and the commutator subgroup......Page 145
5.4. Semidirect products and short exact sequences......Page 147
5.5. Direct sums and wreath products......Page 149
5.6.1. Group actions......Page 150
5.6.2. Linear actions......Page 154
5.6.3. Lie groups......Page 155
5.6.4. Haar measure and lattices......Page 158
5.7. Zariski topology and algebraic groups......Page 161
5.8.1. ��-complexes......Page 168
5.8.2. Borel and Haefliger constructions......Page 169
5.8.3. Groups of finite type......Page 180
5.9.1. Group rings and modules......Page 181
5.9.2. Group cohomology......Page 182
5.9.3. Bounded cohomology of groups......Page 186
5.9.4. Ring derivations......Page 187
5.9.5. Derivations and split extensions......Page 189
5.9.6. Central coextensions and second cohomology......Page 192
6.1. Median spaces......Page 196
6.1.1. A review of median algebras......Page 197
6.1.2. Convexity......Page 198
6.1.3. Examples of median metric spaces......Page 199
6.1.4. Convexity and gate property in median spaces......Page 201
6.1.5. Rectangles and parallel pairs......Page 203
6.1.6. Approximate geodesics and medians; completions of median spaces......Page 206
6.2.1. Definition and basic properties......Page 207
6.2.2. Relationship between median spaces and spaces with measured walls......Page 210
6.2.3. Embedding a space with measured walls in a median space......Page 211
6.2.4. Median spaces have measured walls......Page 213
7.1. Finitely generated groups......Page 220
7.2. Free groups......Page 224
7.3. Presentations of groups......Page 227
7.4. The rank of a free group determines the group. Subgroups......Page 233
7.5.1. Amalgams......Page 234
7.5.2. Graphs of groups......Page 235
7.5.4. Topological interpretation of graphs of groups......Page 237
7.5.5. Constructing finite index subgroups......Page 238
7.5.6. Graphs of groups and group actions on trees......Page 240
7.6. Ping-pong lemma. Examples of free groups......Page 243
7.8. Ping-pong on projective spaces......Page 247
7.9. Cayley graphs......Page 248
7.10.1. Simplicial, cellular and combinatorial volumes of maps......Page 256
7.10.3. Presentations of central coextensions......Page 257
7.10.4. Dehn function and van Kampen diagrams......Page 259
7.11. Residual finiteness......Page 265
7.12. Hopfian and co-hopfian properties......Page 268
7.13. Algorithmic problems in the combinatorial group theory......Page 269
8.1. Quasiisometry......Page 272
8.2. Group-theoretic examples of quasiisometries......Page 282
8.3. A metric version of the Milnor–Schwarz Theorem......Page 288
8.4. Topological coupling......Page 290
8.5. Quasiactions......Page 292
8.6. Quasiisometric rigidity problems......Page 295
8.7. The growth function......Page 296
8.8. Codimension one isoperimetric inequalities......Page 302
8.9. Distortion of a subgroup in a group......Page 304
9.1.1. The number of ends......Page 308
9.1.2. The space of ends......Page 311
9.1.3. Ends of groups......Page 316
9.2.1. Rips complexes......Page 318
9.2.2. Direct system of Rips complexes and coarse homotopy......Page 320
9.3. Metric cell complexes......Page 321
9.4. Connectivity and coarse connectivity......Page 327
9.5. Retractions......Page 333
9.6. Poincaré duality and coarse separation......Page 335
9.7. Metric filling functions......Page 338
9.7.1. Coarse isoperimetric functions and coarse filling radius......Page 339
9.7.2. Quasiisometric invariance of coarse filling functions......Page 341
9.7.3. Higher Dehn functions......Page 346
9.7.4. Coarse Besikovitch inequality......Page 351
10.1. The Axiom of Choice and its weaker versions......Page 354
10.2. Ultrafilters and the Stone–Čech compactification......Page 360
10.3. Elements of non-standard algebra......Page 361
10.4. Ultralimits of families of metric spaces......Page 365
10.5. Completeness of ultralimits and incompleteness of ultrafilters......Page 369
10.6. Asymptotic cones of metric spaces......Page 373
10.7. Ultralimits of asymptotic cones are asymptotic cones......Page 377
10.8. Asymptotic cones and quasiisometries......Page 379
10.9. Assouad-type theorems......Page 381
11.1. Hyperbolicity according to Rips......Page 384
11.2. Geometry and topology of real trees......Page 388
11.3. Gromov hyperbolicity......Page 389
11.4. Ultralimits and stability of geodesics in Rips-hyperbolic spaces......Page 393
11.5. Local geodesics in hyperbolic spaces......Page 397
11.6. Quasiconvexity in hyperbolic spaces......Page 400
11.7. Nearest-point projections......Page 402
11.8. Geometry of triangles in Rips-hyperbolic spaces......Page 403
11.9. Divergence of geodesics in hyperbolic metric spaces......Page 406
11.10. Morse Lemma revisited......Page 408
11.11. Ideal boundaries......Page 411
11.12. Gromov bordification of Gromov-hyperbolic spaces......Page 419
11.13.1. Extended Morse Lemma......Page 423
11.13.2. The extension theorem......Page 425
11.13.3. Boundary extension and quasiactions......Page 427
11.14. Hyperbolic groups......Page 428
11.15. Ideal boundaries of hyperbolic groups......Page 431
11.16. Linear isoperimetric inequality and Dehn algorithm for hyperbolic groups......Page 435
11.17. The small cancellation theory......Page 438
11.18. The Rips construction......Page 439
11.19. Central coextensions of hyperbolic groups and quasiisometries......Page 440
11.20. Characterization of hyperbolicity using asymptotic cones......Page 444
11.21.1. The minsize......Page 450
11.21.2. The constriction......Page 451
11.22. Filling invariants of hyperbolic spaces......Page 453
11.22.1. Filling area......Page 454
11.22.2. Filling radius......Page 455
11.22.3. Orders of Dehn functions of non-hyperbolic groups and higher Dehn functions......Page 458
11.23. Asymptotic cones, actions on trees and isometric actions on hyperbolic spaces......Page 459
11.24. Summary of equivalent definitions of hyperbolicity......Page 462
11.25. Further properties of hyperbolic groups......Page 463
11.26. Relatively hyperbolic spaces and groups......Page 466
12.1. Semisimple Lie groups and their symmetric spaces......Page 470
12.2. Lattices......Page 472
12.3. Examples of lattices......Page 473
12.4. Rigidity and superrigidity......Page 475
12.6.1. Zariski density......Page 477
12.6.2. Parabolic elements and non-compactness......Page 479
12.6.3. Thick-thin decomposition......Page 481
12.7. Central coextensions......Page 483
13.1. Free abelian groups......Page 486
13.2. Classification of finitely generated abelian groups......Page 489
13.3. Automorphisms of \Zⁿ......Page 492
13.4. Nilpotent groups......Page 495
13.5. Polycyclic groups......Page 505
13.6. Solvable groups: Definition and basic properties......Page 510
13.7. Free solvable groups and the Magnus embedding......Page 512
13.8. Solvable versus polycyclic......Page 514
14.1. Wolf’s Theorem for semidirect products \Zⁿ⋊\Z......Page 518
14.1.1. Geometry of ��₃(\Z)......Page 520
14.1.2. Distortion of subgroups of solvable groups......Page 524
14.1.3. Distortion of subgroups in nilpotent groups......Page 526
14.2. Polynomial growth of nilpotent groups......Page 535
14.3. Wolf’s Theorem......Page 536
14.4. Milnor’s Theorem......Page 538
14.5. Failure of QI rigidity for solvable groups......Page 541
14.6. Virtually nilpotent subgroups of ����(��)......Page 542
14.7.1. Some useful linear algebra......Page 545
14.7.2. Zassenhaus neighborhoods......Page 546
14.7.3. Jordan’s Theorem......Page 549
14.8. Virtually solvable subgroups of ����(��,\C)......Page 551
Chapter 15. The Tits Alternative......Page 558
15.1. Outline of the proof......Page 559
15.2. Separating sets......Page 561
15.4. Existence of very proximal elements: Proof of Theorem 15.6......Page 562
15.4.1. Proximality criteria......Page 563
15.4.2. Constructing very proximal elements......Page 564
15.5. Finding ping-pong partners: Proof of Theorem 15.7......Page 566
15.6. The Tits Alternative without finite generation assumption......Page 567
15.7. Groups satisfying the Tits Alternative......Page 568
16.1. Topological transformation groups......Page 570
16.2. Regular Growth Theorem......Page 572
16.3. Consequences of the Regular Growth Theorem......Page 576
16.4. Weakly polynomial growth......Page 577
16.5. Displacement function......Page 578
16.6. Proof of Gromov’s Theorem......Page 579
16.7. Quasiisometric rigidity of nilpotent and abelian groups......Page 582
16.8. Further developments......Page 583
17.1. Paradoxical decompositions......Page 586
17.2. Step 1: A paradoxical decomposition of the free group ��₂......Page 589
17.3. Step 2: The Hausdorff Paradox......Page 590
17.4. Step 3: Spheres of dimension \gq2 are paradoxical......Page 591
17.5. Step 4: Euclidean unit balls are paradoxical......Page 592
18.1. Amenable graphs......Page 594
18.2. Amenability and quasiisometry......Page 599
18.3. Amenability of groups......Page 604
18.4. Følner Property......Page 609
18.5. Amenability, paradoxality and the Følner Property......Page 613
18.6. Supramenability and weakly paradoxical actions......Page 617
18.7. Quantitative approaches to non-amenability and weak paradoxality......Page 622
18.8. Uniform amenability and ultrapowers......Page 627
18.9. Quantitative approaches to amenability......Page 629
18.10. Summary of equivalent definitions of amenability......Page 633
18.11. Amenable hierarchy......Page 634
19.1. Classes of Banach spaces stable with respect to ultralimits......Page 636
19.2. Limit actions and point-selection theorem......Page 641
19.3. Properties for actions on Hilbert spaces......Page 646
19.4. Kazhdan’s Property (T) and the Haagerup Property......Page 648
19.5. Groups acting non-trivially on trees do not have Property (T)......Page 654
19.6. Property FH, a-T-menability, and group actions on median spaces......Page 657
19.7. Fixed-point property and proper actions for ��^{��}-spaces......Page 660
19.8. Groups satisfying Property (T) and the spectral gap......Page 662
19.9. Failure of quasiisometric invariance of Property (T)......Page 664
19.10. Summary of examples......Page 665
20.1. Maps to trees and hyperbolic metrics on 2-dimensional simplicial complexes......Page 668
20.2. Transversal graphs and Dunwoody tracks......Page 673
20.3. Existence of minimal Dunwoody tracks......Page 677
20.4.1. Stationarity......Page 680
20.4.2. Disjointness of essential minimal tracks......Page 682
20.5. The Stallings Theorem for almost finitely presented groups......Page 685
20.6. Accessibility......Page 687
20.7. QI rigidity of virtually free groups and free products......Page 692
Chapter 21. Proof of Stallings’ Theorem using harmonic functions......Page 696
21.1. Proof of Stallings’ Theorem......Page 698
21.2. Non-amenability......Page 702
21.3. An existence theorem for harmonic functions......Page 704
21.4. Energy of minimum and maximum of two smooth functions......Page 707
21.5.1. Positive energy gap implies existence of an energy minimizer......Page 708
21.5.2. Some coarea estimates......Page 711
21.5.3. Energy comparison in the case of a linear isoperimetric inequality......Page 713
21.5.4. Proof of positivity of the energy gap......Page 715
Chapter 22. Quasiconformal mappings......Page 718
22.1. Linear algebra and eccentricity of ellipsoids......Page 719
22.2. Quasisymmetric maps......Page 720
22.3. Quasiconformal maps......Page 722
22.4.1. Some notions and results from real analysis......Page 723
22.4.2. Differentiability properties of quasiconformal mappings......Page 726
22.5. Quasisymmetric maps and hyperbolic geometry......Page 733
Chapter 23. Groups quasiisometric to ℊⁿ......Page 738
23.1. Uniformly quasiconformal groups......Page 739
23.2. Hyperbolic extension of uniformly quasiconformal groups......Page 740
23.3. Least volume ellipsoids......Page 741
23.4. Invariant measurable conformal structure......Page 742
23.5.1. Beltrami equation......Page 745
23.5.2. Measurable Riemannian metrics......Page 746
23.6. Proof of Tukia’s Theorem on uniformly quasiconformal groups......Page 747
23.7. QI rigidity for surface groups......Page 750
Chapter 24. Quasiisometries of non-uniform lattices in ℊⁿ......Page 754
24.1. Coarse topology of truncated hyperbolic spaces......Page 755
24.2. Hyperbolic extension......Page 759
24.3. Mostow Rigidity Theorem......Page 760
24.4. Zooming in......Page 764
24.5. Inverted linear mappings......Page 766
24.6. Scattering......Page 769
24.7. Schwartz Rigidity Theorem......Page 771
25.1.1. Uniform lattices......Page 774
25.1.2. Non-uniform lattices......Page 775
25.1.3. Symmetric spaces with Euclidean de Rham factors and Lie groups with nilpotent normal subgroups......Page 777
25.1.4. QI rigidity for hyperbolic spaces and groups......Page 778
25.1.5. Failure of QI rigidity......Page 781
25.2. Rigidity of relatively hyperbolic groups......Page 783
25.3. Rigidity of classes of amenable groups......Page 785
25.4. Bi-Lipschitz vs. quasiisometric......Page 788
25.5. Various other QI rigidity results and problems......Page 790
26.1. Introduction......Page 798
26.3. Platonov’s Theorem......Page 799
26.4. Proof of Platonov’s Theorem......Page 801
26.5. The Idempotent Conjecture for linear groups......Page 803
26.6. Proof of Formanek’s criterion......Page 804
26.7. Notes......Page 806
Bibliography......Page 808
Index......Page 834
Back Cover......Page 841