**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

- Chapter 13. Solvable groups 465486
- 13.1. Free abelian groups 465486
- 13.2. Classification of finitely generated abelian groups 468489
- 13.3. Automorphisms of \Zⁿ 471492
- 13.4. Nilpotent groups 474495
- 13.5. Polycyclic groups 484505
- 13.6. Solvable groups: Definition and basic properties 489510
- 13.7. Free solvable groups and the Magnus embedding 491512
- 13.8. Solvable versus polycyclic 493514

- Chapter 14. Geometric aspects of solvable groups 497518
- 14.1. Wolf’s Theorem for semidirect products \Zⁿ⋊\Z 497518
- 14.2. Polynomial growth of nilpotent groups 514535
- 14.3. Wolf’s Theorem 515536
- 14.4. Milnor’s Theorem 517538
- 14.5. Failure of QI rigidity for solvable groups 520541
- 14.6. Virtually nilpotent subgroups of 𝐺𝐿(𝑛) 521542
- 14.7. Discreteness and nilpotence in Lie groups 524545
- 14.8. Virtually solvable subgroups of 𝐺𝐿(𝑛,\C) 530551

- Chapter 15. The Tits Alternative 537558
- 15.1. Outline of the proof 538559
- 15.2. Separating sets 540561
- 15.3. Proof of the existence of free subsemigroups 541562
- 15.4. Existence of very proximal elements: Proof of Theorem 15.6 541562
- 15.5. Finding ping-pong partners: Proof of Theorem 15.7 545566
- 15.6. The Tits Alternative without finite generation assumption 546567
- 15.7. Groups satisfying the Tits Alternative 547568

- Chapter 16. Gromov’s Theorem 549570
- 16.1. Topological transformation groups 549570
- 16.2. Regular Growth Theorem 551572
- 16.3. Consequences of the Regular Growth Theorem 555576
- 16.4. Weakly polynomial growth 556577
- 16.5. Displacement function 557578
- 16.6. Proof of Gromov’s Theorem 558579
- 16.7. Quasiisometric rigidity of nilpotent and abelian groups 561582
- 16.8. Further developments 562583

- Chapter 17. The Banach–Tarski Paradox 565586
- Chapter 18. Amenability and paradoxical decomposition 573594
- 18.1. Amenable graphs 573594
- 18.2. Amenability and quasiisometry 578599
- 18.3. Amenability of groups 583604
- 18.4. Følner Property 588609
- 18.5. Amenability, paradoxality and the Følner Property 592613
- 18.6. Supramenability and weakly paradoxical actions 596617
- 18.7. Quantitative approaches to non-amenability and weak paradoxality 601622
- 18.8. Uniform amenability and ultrapowers 606627
- 18.9. Quantitative approaches to amenability 608629
- 18.10. Summary of equivalent definitions of amenability 612633
- 18.11. Amenable hierarchy 613634

- Chapter 19. Ultralimits, fixed-point properties, proper actions 615636
- 19.1. Classes of Banach spaces stable with respect to ultralimits 615636
- 19.2. Limit actions and point-selection theorem 620641
- 19.3. Properties for actions on Hilbert spaces 625646
- 19.4. Kazhdan’s Property (T) and the Haagerup Property 627648
- 19.5. Groups acting non-trivially on trees do not have Property (T) 633654
- 19.6. Property FH, a-T-menability, and group actions on median spaces 636657
- 19.7. Fixed-point property and proper actions for 𝐿^{𝑝}-spaces 639660
- 19.8. Groups satisfying Property (T) and the spectral gap 641662
- 19.9. Failure of quasiisometric invariance of Property (T) 643664
- 19.10. Summary of examples 644665

- Chapter 20. The Stallings Theorem and accessibility 647668
- 20.1. Maps to trees and hyperbolic metrics on 2-dimensional simplicial complexes 647668
- 20.2. Transversal graphs and Dunwoody tracks 652673
- 20.3. Existence of minimal Dunwoody tracks 656677
- 20.4. Properties of minimal tracks 659680
- 20.5. The Stallings Theorem for almost finitely presented groups 664685
- 20.6. Accessibility 666687
- 20.7. QI rigidity of virtually free groups and free products 671692

- Chapter 21. Proof of Stallings’ Theorem using harmonic functions 675696
- Chapter 22. Quasiconformal mappings 697718
- Chapter 23. Groups quasiisometric to ℍⁿ 717738
- 23.1. Uniformly quasiconformal groups 718739
- 23.2. Hyperbolic extension of uniformly quasiconformal groups 719740
- 23.3. Least volume ellipsoids 720741
- 23.4. Invariant measurable conformal structure 721742
- 23.5. Quasiconformality in dimension 2 724745
- 23.6. Proof of Tukia’s Theorem on uniformly quasiconformal groups 726747
- 23.7. QI rigidity for surface groups 729750

- Chapter 24. Quasiisometries of non-uniform lattices in ℍⁿ 733754
- Chapter 25. A survey of quasiisometric rigidity 753774
- 25.1. Rigidity of symmetric spaces, lattices and hyperbolic groups 753774
- 25.1.1. Uniform lattices 753774
- 25.1.2. Non-uniform lattices 754775
- 25.1.3. Symmetric spaces with Euclidean de Rham factors and Lie groups with nilpotent normal subgroups 756777
- 25.1.4. QI rigidity for hyperbolic spaces and groups 757778
- 25.1.5. Failure of QI rigidity 760781
- 25.1.6. Rigidity of random groups 762783

- 25.2. Rigidity of relatively hyperbolic groups 762783
- 25.3. Rigidity of classes of amenable groups 764785
- 25.4. Bi-Lipschitz vs. quasiisometric 767788
- 25.5. Various other QI rigidity results and problems 769790

- Chapter 26. Appendix by Bogdan Nica: Three theorems on linear groups 777798
- Bibliography 787808
- Index 813834
- Back Cover Back Cover1841

Table of Contents pages: 1 2