Geometric Group Theory

Description

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