**Colloquium Publications**

Volume: 63;
2018;
819 pp;
Hardcover

MSC: Primary 20; 57;

**Print ISBN: 978-1-4704-1104-6
Product Code: COLL/63**

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

**Electronic ISBN: 978-1-4704-4164-7
Product Code: COLL/63.E**

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

#### Supplemental Materials

# Geometric Group Theory

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*Cornelia Druţu; Michael Kapovich*

With an appendix by Bogdan Nica

The key idea in geometric group theory is to
study infinite groups by endowing them with a metric and treating them
as geometric spaces. This applies to many groups naturally appearing
in topology, geometry, and algebra, such as fundamental groups of
manifolds, groups of matrices with integer coefficients, etc. The
primary focus of this book is to cover the foundations of
geometric group theory, including coarse topology, ultralimits and
asymptotic cones, hyperbolic groups, isoperimetric inequalities,
growth of groups, amenability, Kazhdan's Property (T) and the Haagerup
property, as well as their characterizations in terms of group actions
on median spaces and spaces with walls.

The book contains proofs of several fundamental results of
geometric group theory, such as Gromov's theorem on groups of
polynomial growth, Tits's alternative, Stallings's theorem on ends of
groups, Dunwoody's accessibility theorem, the Mostow Rigidity Theorem,
and quasiisometric rigidity theorems of Tukia and Schwartz. This is
the first book in which geometric group theory is presented in a form
accessible to advanced graduate students and young research
mathematicians. It fills a big gap in the literature and will be used
by researchers in geometric group theory and its applications.

#### Readership

Graduate students and researchers interested in geometric group theory.

#### Reviews & Endorsements

[This book] offers a comprehensive account of major developments in geometric group theory in the 20th century. It is inevitable that some topics are mentioned only briefly, which is compensated by the extensive bibliography of over 600 references, both old and recent.

-- Igor Belegradek, Mathematical Reviews

It will undoubtedly be an essential reference for those working in GGT and related areas in differential geometry and topology. Its wealth of immensely valuable historical commentary is particularly appreciated.

-- Scott Taylor, MAA Reviews

#### Table of Contents

# Table of Contents

## Geometric Group Theory

Table of Contents pages: 1 2

- Cover Cover11
- Title page i2
- Contents v6
- Preface xiii14
- Chapter 1. Geometry and topology 122
- Chapter 2. Metric spaces 2344
- 2.1. General metric spaces 2344
- 2.2. Length metric spaces 2546
- 2.3. Graphs as length spaces 2748
- 2.4. Hausdorff and Gromov–Hausdorff distances. Nets 2849
- 2.5. Lipschitz maps and Banach–Mazur distance 3051
- 2.6. Hausdorff dimension 3455
- 2.7. Norms and valuations 3556
- 2.8. Norms on field extensions. Adeles 3960
- 2.9. Metrics on affine and projective spaces 4364
- 2.10. Quasiprojective transformations. Proximal transformations 4869
- 2.11. Kernels and distance functions 5172

- Chapter 3. Differential geometry 5980
- 3.1. Smooth manifolds 5980
- 3.2. Smooth partition of unity 6182
- 3.3. Riemannian metrics 6182
- 3.4. Riemannian volume 6485
- 3.5. Volume growth and isoperimetric functions. Cheeger constant 6889
- 3.6. Curvature 7192
- 3.7. Riemannian manifolds of bounded geometry 7394
- 3.8. Metric simplicial complexes of bounded geometry and systolic inequalities 7495
- 3.9. Harmonic functions 79100
- 3.10. Spectral interpretation of the Cheeger constant 82103
- 3.11. Comparison geometry 82103

- Chapter 4. Hyperbolic space 91112
- 4.1. Moebius transformations 91112
- 4.2. Real-hyperbolic space 94115
- 4.3. Classification of isometries 99120
- 4.4. Hyperbolic trigonometry 102123
- 4.5. Triangles and curvature of ℍⁿ 105126
- 4.6. Distance function on ℍⁿ 108129
- 4.7. Hyperbolic balls and spheres 110131
- 4.8. Horoballs and horospheres in \bHⁿ 110131
- 4.9. \bHⁿ as a symmetric space 112133
- 4.10. Inscribed radius and thinness of hyperbolic triangles 116137
- 4.11. Existence-uniqueness theorem for triangles 118139

- Chapter 5. Groups and their actions 119140
- 5.1. Subgroups 120141
- 5.2. Virtual isomorphisms of groups and commensurators 122143
- 5.3. Commutators and the commutator subgroup 124145
- 5.4. Semidirect products and short exact sequences 126147
- 5.5. Direct sums and wreath products 128149
- 5.6. Geometry of group actions 129150
- 5.7. Zariski topology and algebraic groups 140161
- 5.8. Group actions on complexes 147168
- 5.9. Cohomology 160181

- Chapter 6. Median spaces and spaces with measured walls 175196
- Chapter 7. Finitely generated and finitely presented groups 199220
- 7.1. Finitely generated groups 199220
- 7.2. Free groups 203224
- 7.3. Presentations of groups 206227
- 7.4. The rank of a free group determines the group. Subgroups 212233
- 7.5. Free constructions: Amalgams of groups and graphs of groups 213234
- 7.6. Ping-pong lemma. Examples of free groups 222243
- 7.7. Free subgroups in 𝑆𝑈(2) 226247
- 7.8. Ping-pong on projective spaces 226247
- 7.9. Cayley graphs 227248
- 7.10. Volumes of maps of cell complexes and Van Kampen diagrams 235256
- 7.11. Residual finiteness 244265
- 7.12. Hopfian and co-hopfian properties 247268
- 7.13. Algorithmic problems in the combinatorial group theory 248269

- Chapter 8. Coarse geometry 251272
- 8.1. Quasiisometry 251272
- 8.2. Group-theoretic examples of quasiisometries 261282
- 8.3. A metric version of the Milnor–Schwarz Theorem 267288
- 8.4. Topological coupling 269290
- 8.5. Quasiactions 271292
- 8.6. Quasiisometric rigidity problems 274295
- 8.7. The growth function 275296
- 8.8. Codimension one isoperimetric inequalities 281302
- 8.9. Distortion of a subgroup in a group 283304

- Chapter 9. Coarse topology 287308
- Chapter 10. Ultralimits of metric spaces 333354
- 10.1. The Axiom of Choice and its weaker versions 333354
- 10.2. Ultrafilters and the Stone–Čech compactification 339360
- 10.3. Elements of non-standard algebra 340361
- 10.4. Ultralimits of families of metric spaces 344365
- 10.5. Completeness of ultralimits and incompleteness of ultrafilters 348369
- 10.6. Asymptotic cones of metric spaces 352373
- 10.7. Ultralimits of asymptotic cones are asymptotic cones 356377
- 10.8. Asymptotic cones and quasiisometries 358379
- 10.9. Assouad-type theorems 360381

- Chapter 11. Gromov-hyperbolic spaces and groups 363384
- 11.1. Hyperbolicity according to Rips 363384
- 11.2. Geometry and topology of real trees 367388
- 11.3. Gromov hyperbolicity 368389
- 11.4. Ultralimits and stability of geodesics in Rips-hyperbolic spaces 372393
- 11.5. Local geodesics in hyperbolic spaces 376397
- 11.6. Quasiconvexity in hyperbolic spaces 379400
- 11.7. Nearest-point projections 381402
- 11.8. Geometry of triangles in Rips-hyperbolic spaces 382403
- 11.9. Divergence of geodesics in hyperbolic metric spaces 385406
- 11.10. Morse Lemma revisited 387408
- 11.11. Ideal boundaries 390411
- 11.12. Gromov bordification of Gromov-hyperbolic spaces 398419
- 11.13. Boundary extension of quasiisometries of hyperbolic spaces 402423
- 11.14. Hyperbolic groups 407428
- 11.15. Ideal boundaries of hyperbolic groups 410431
- 11.16. Linear isoperimetric inequality and Dehn algorithm for hyperbolic groups 414435
- 11.17. The small cancellation theory 417438
- 11.18. The Rips construction 418439
- 11.19. Central coextensions of hyperbolic groups and quasiisometries 419440
- 11.20. Characterization of hyperbolicity using asymptotic cones 423444
- 11.21. Size of loops 429450
- 11.22. Filling invariants of hyperbolic spaces 432453
- 11.23. Asymptotic cones, actions on trees and isometric actions on hyperbolic spaces 438459
- 11.24. Summary of equivalent definitions of hyperbolicity 441462
- 11.25. Further properties of hyperbolic groups 442463
- 11.26. Relatively hyperbolic spaces and groups 445466

- Chapter 12. Lattices in Lie groups 449470

Table of Contents pages: 1 2