8.8. Codimension one isoperimetric inequalities Distortion of a subgroup in a group 283

Transcription

1 Contents Preface xiii Chapter 1. Geometry and topology Set-theoretic preliminaries General notation Growth rates of functions Jensen s inequality Measure and integral Measures Finitely additive integrals Topological spaces. Lebesgue covering dimension Exhaustions of locally compact spaces Direct and inverse limits Graphs Complexes and homology Simplicial complexes Cell complexes 19 Chapter 2. Metric spaces General metric spaces Length metric spaces Graphs as length spaces Hausdorff and Gromov Hausdorff distances. Nets Lipschitz maps and Banach Mazur distance Lipschitz and locally Lipschitz maps Bi-Lipschitz maps. The Banach Mazur distance Hausdorff dimension Norms and valuations Norms on field extensions. Adeles Metrics on affine and projective spaces Quasiprojective transformations. Proximal transformations Kernels and distance functions 51 Chapter 3. Differential geometry Smooth manifolds Smooth partition of unity Riemannian metrics Riemannian volume Volume growth and isoperimetric functions. Cheeger constant Curvature 71 v

2 vi CONTENTS 3.7. Riemannian manifolds of bounded geometry Metric simplicial complexes of bounded geometry and systolic inequalities Harmonic functions Spectral interpretation of the Cheeger constant Comparison geometry Alexandrov curvature and CAT(κ) spaces Cartan s Fixed-Point Theorem Ideal boundary, horoballs and horospheres 88 Chapter 4. Hyperbolic space Moebius transformations Real-hyperbolic space Classification of isometries Hyperbolic trigonometry Triangles and curvature of H n Distance function on H n Hyperbolic balls and spheres Horoballs and horospheres in H n H n as a symmetric space Inscribed radius and thinness of hyperbolic triangles Existence-uniqueness theorem for triangles 118 Chapter 5. Groups and their actions Subgroups Virtual isomorphisms of groups and commensurators Commutators and the commutator subgroup Semidirect products and short exact sequences Direct sums and wreath products Geometry of group actions Group actions Linear actions Lie groups Haar measure and lattices Geometric actions Zariski topology and algebraic groups Group actions on complexes G-complexes Borel and Haefliger constructions Groups of finite type Cohomology Group rings and modules Group cohomology Bounded cohomology of groups Ring derivations Derivations and split extensions Central coextensions and second cohomology 171

3 CONTENTS vii Chapter 6. Median spaces and spaces with measured walls Median spaces A review of median algebras Convexity Examples of median metric spaces Convexity and gate property in median spaces Rectangles and parallel pairs Approximate geodesics and medians; completions of median spaces Spaces with measured walls Definition and basic properties Relationship between median spaces and spaces with measured walls Embedding a space with measured walls in a median space Median spaces have measured walls 192 Chapter 7. Finitely generated and finitely presented groups Finitely generated groups Free groups Presentations of groups The rank of a free group determines the group. Subgroups Free constructions: Amalgams of groups and graphs of groups Amalgams Graphs of groups Converting graphs of groups into amalgams Topological interpretation of graphs of groups Constructing finite index subgroups Graphs of groups and group actions on trees Ping-pong lemma. Examples of free groups Free subgroups in SU(2) Ping-pong on projective spaces Cayley graphs Volumes of maps of cell complexes and Van Kampen diagrams Simplicial, cellular and combinatorial volumes of maps Topological interpretation of finite-presentability Presentations of central coextensions Dehn function and van Kampen diagrams Residual finiteness Hopfian and co-hopfian properties Algorithmic problems in the combinatorial group theory 248 Chapter 8. Coarse geometry Quasiisometry Group-theoretic examples of quasiisometries A metric version of the Milnor Schwarz Theorem Topological coupling Quasiactions Quasiisometric rigidity problems The growth function 275

4 viii CONTENTS 8.8. Codimension one isoperimetric inequalities Distortion of a subgroup in a group 283 Chapter 9. Coarse topology Ends The number of ends The space of ends Ends of groups Rips complexes and coarse homotopy theory Rips complexes Direct system of Rips complexes and coarse homotopy Metric cell complexes Connectivity and coarse connectivity Retractions Poincaré duality and coarse separation Metric filling functions Coarse isoperimetric functions and coarse filling radius Quasiisometric invariance of coarse filling functions Higher Dehn functions Coarse Besikovitch inequality 330 Chapter 10. Ultralimits of metric spaces The Axiom of Choice and its weaker versions Ultrafilters and the Stone Čech compactification Elements of non-standard algebra Ultralimits of families of metric spaces Completeness of ultralimits and incompleteness of ultrafilters Asymptotic cones of metric spaces Ultralimits of asymptotic cones are asymptotic cones Asymptotic cones and quasiisometries Assouad-type theorems 360 Chapter 11. Gromov-hyperbolic spaces and groups Hyperbolicity according to Rips Geometry and topology of real trees Gromov hyperbolicity Ultralimits and stability of geodesics in Rips-hyperbolic spaces Local geodesics in hyperbolic spaces Quasiconvexity in hyperbolic spaces Nearest-point projections Geometry of triangles in Rips-hyperbolic spaces Divergence of geodesics in hyperbolic metric spaces Morse Lemma revisited Ideal boundaries Gromov bordification of Gromov-hyperbolic spaces Boundary extension of quasiisometries of hyperbolic spaces Extended Morse Lemma The extension theorem Boundary extension and quasiactions 406

5 CONTENTS ix Conical limit points of quasiactions Hyperbolic groups Ideal boundaries of hyperbolic groups Linear isoperimetric inequality and Dehn algorithm for hyperbolic groups The small cancellation theory The Rips construction Central coextensions of hyperbolic groups and quasiisometries Characterization of hyperbolicity using asymptotic cones Size of loops The minsize The constriction Filling invariants of hyperbolic spaces Filling area Filling radius Orders of Dehn functions of non-hyperbolic groups and higher Dehn functions Asymptotic cones, actions on trees and isometric actions on hyperbolic spaces Summary of equivalent definitions of hyperbolicity Further properties of hyperbolic groups Relatively hyperbolic spaces and groups 445 Chapter 12. Lattices in Lie groups Semisimple Lie groups and their symmetric spaces Lattices Examples of lattices Rigidity and superrigidity Commensurators of lattices Lattices in PO(n, 1) Zariski density Parabolic elements and non-compactness Thick-thin decomposition Central coextensions 462 Chapter 13. Solvable groups Free abelian groups Classification of finitely generated abelian groups Automorphisms of Z n Nilpotent groups Polycyclic groups Solvable groups: Definition and basic properties Free solvable groups and the Magnus embedding Solvable versus polycyclic 493 Chapter 14. Geometric aspects of solvable groups Wolf s Theorem for semidirect products Z n Z Geometry of H 3 (Z) Distortion of subgroups of solvable groups 503

6 x CONTENTS Distortion of subgroups in nilpotent groups Polynomial growth of nilpotent groups Wolf s Theorem Milnor s Theorem Failure of QI rigidity for solvable groups Virtually nilpotent subgroups of GL(n) Discreteness and nilpotence in Lie groups Some useful linear algebra Zassenhaus neighborhoods Jordan s Theorem Virtually solvable subgroups of GL(n, C) 530 Chapter 15. The Tits Alternative Outline of the proof Separating sets Proof of the existence of free subsemigroups Existence of very proximal elements: Proof of Theorem Proximality criteria Constructing very proximal elements Finding ping-pong partners: Proof of Theorem The Tits Alternative without finite generation assumption Groups satisfying the Tits Alternative 547 Chapter 16. Gromov s Theorem Topological transformation groups Regular Growth Theorem Consequences of the Regular Growth Theorem Weakly polynomial growth Displacement function Proof of Gromov s Theorem Quasiisometric rigidity of nilpotent and abelian groups Further developments 562 Chapter 17. The Banach Tarski Paradox Paradoxical decompositions Step 1: A paradoxical decomposition of the free group F Step 2: The Hausdorff Paradox Step 3: Spheres of dimension 2 are paradoxical Step 4: Euclidean unit balls are paradoxical 571 Chapter 18. Amenability and paradoxical decomposition Amenable graphs Amenability and quasiisometry Amenability of groups Følner Property Amenability, paradoxality and the Følner Property Supramenability and weakly paradoxical actions Quantitative approaches to non-amenability and weak paradoxality Uniform amenability and ultrapowers 606

7 CONTENTS xi Quantitative approaches to amenability Summary of equivalent definitions of amenability Amenable hierarchy 613 Chapter 19. Ultralimits, fixed-point properties, proper actions Classes of Banach spaces stable with respect to ultralimits Limit actions and point-selection theorem Properties for actions on Hilbert spaces Kazhdan s Property (T) and the Haagerup Property Groups acting non-trivially on trees do not have Property (T) Property FH, a-t-menability, and group actions on median spaces Fixed-point property and proper actions for L p -spaces Groups satisfying Property (T) and the spectral gap Failure of quasiisometric invariance of Property (T) Summary of examples 644 Chapter 20. The Stallings Theorem and accessibility Maps to trees and hyperbolic metrics on 2-dimensional simplicial complexes Transversal graphs and Dunwoody tracks Existence of minimal Dunwoody tracks Properties of minimal tracks Stationarity Disjointness of essential minimal tracks The Stallings Theorem for almost finitely presented groups Accessibility QI rigidity of virtually free groups and free products 671 Chapter 21. Proof of Stallings Theorem using harmonic functions Proof of Stallings Theorem Non-amenability An existence theorem for harmonic functions Energy of minimum and maximum of two smooth functions A compactness theorem for harmonic functions Positive energy gap implies existence of an energy minimizer Some coarea estimates Energy comparison in the case of a linear isoperimetric inequality Proof of positivity of the energy gap 694 Chapter 22. Quasiconformal mappings Linear algebra and eccentricity of ellipsoids Quasisymmetric maps Quasiconformal maps Analytical properties of quasiconformal mappings Some notions and results from real analysis Differentiability properties of quasiconformal mappings Quasisymmetric maps and hyperbolic geometry 712

8 xii CONTENTS Chapter 23. Groups quasiisometric to H n Uniformly quasiconformal groups Hyperbolic extension of uniformly quasiconformal groups Least volume ellipsoids Invariant measurable conformal structure Quasiconformality in dimension Beltrami equation Measurable Riemannian metrics Proof of Tukia s Theorem on uniformly quasiconformal groups QI rigidity for surface groups 729 Chapter 24. Quasiisometries of non-uniform lattices in H n Coarse topology of truncated hyperbolic spaces Hyperbolic extension Mostow Rigidity Theorem Zooming in Inverted linear mappings Scattering Schwartz Rigidity Theorem 750 Chapter 25. A survey of quasiisometric rigidity Rigidity of symmetric spaces, lattices and hyperbolic groups Uniform lattices Non-uniform lattices Symmetric spaces with Euclidean de Rham factors and Lie groups with nilpotent normal subgroups QI rigidity for hyperbolic spaces and groups Failure of QI rigidity Rigidity of random groups Rigidity of relatively hyperbolic groups Rigidity of classes of amenable groups Bi-Lipschitz vs. quasiisometric Various other QI rigidity results and problems 769 Chapter 26. Appendix by Bogdan Nica: Three theorems on linear groups Introduction Virtual and residual properties of groups Platonov s Theorem Proof of Platonov s Theorem The Idempotent Conjecture for linear groups Proof of Formanek s criterion Notes 785 Bibliography 787 Index 813