Topics in Groups and Geometry: Growth, Amenability, and Random Walks

Description

This book provides a detailed exposition of a wide range of topics in geometric group theory, inspired by Gromov’s pivotal work in the 1980s. It includes classical theorems on nilpotent groups and solvable groups, a fundamental study of the growth of groups, a detailed look at asymptotic cones, and a discussion of related subjects including filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups. The results are unified under the common theme of Gromov’s theorem, namely that finitely generated groups of polynomial growth are virtually nilpotent. This beautiful result gave birth to a fascinating new area of research which is still active today. The purpose of the book is to collect these naturally related results together in one place, most of which are scattered throughout the literature, some of them appearing here in book form for the first time. In this way, the connections between these topics are revealed, providing a pleasant introduction to geometric group theory based on ideas surrounding Gromov's theorem. The book will be of interest to mature undergraduate and graduate students in mathematics who are familiar with basic group theory and topology, and who wish to learn more about geometric, analytic, and probabilistic aspects of infinite groups. Foreword Preface Contents Notation Part I Algebraic Theory Chapter 1 Free Groups 1.1 Words 1.2 Definition of Free Groups 1.3 Reduced Forms 1.4 Existence of Free Groups 1.5 Subgroups, Quotients, and Extensions of Finitely Generated Groups 1.6 Subgroups of Free Groups 1.7 The Ping-Pong Lemma 1.8 Free Abelian Groups 1.9 Notes 1.10 Exercises Chapter 2 Nilpotent Groups 2.1 Commutator Identities 2.2 The Lower Central Series 2.3 The Upper Central Series 2.4 Two Examples 2.5 Nilpotent Ideals 2.6 Torsion-Free Finitely Generated Nilpotent Groups 2.7 Finitely Generated Nilpotent Groups with Torsion 2.8 Notes 2.9 Exercises Chapter 3 Residual Finiteness and the Zassenhaus Filtration 3.1 The Lie Ring of a Group 3.2 The Zassenhaus Filtration 3.3 Residually-p and Residually Finite Groups 3.4 The Theorems of Malcev and G. Baumslag 3.5 Residual Finiteness of Free Groups 3.6 Notes 3.7 Exercises Chapter 4 Solvable Groups 4.1 Solvable Groups: Definitions and Relations with Nilpotent Groups 4.2 Two Important Examples: UT(n,R) and B(n,R) 4.3 Statement of Malcev’s Theorem on Solvable Groups 4.4 Wedderburn Theory 4.5 Proof of Malcev’s Theorem on Solvable Groups 4.6 Notes 4.7 Exercises Chapter 5 Polycyclic Groups 5.1 Polycyclic, Polycyclic-by-Finite, and Poly-Infinite-Cyclic Groups 5.2 The Hirsch Number 5.3 Malcev’s Theorem on Polycyclic Groups 5.4 Malcev’s Theorem on Polycyclic-by-Finite Groups 5.5 The Auslander–Swan Theorem 5.6 Notes 5.7 Exercises Chapter 6 The Burnside Problem 6.1 Formulation of the Burnside Problems 6.2 Locally Finite Groups and the General Burnside Problem 6.3 The General Burnside Problem for Polycyclic-by-Finite and Solvable Groups 6.4 The Bounded Burnside Problem for Linear Groups 6.5 The Golod–Shafarevich Construction 6.6 Notes 6.7 Exercises Part II Geometric Theory Chapter 7 Finitely Generated Groups and their Growth Functions 7.1 The Word Metric 7.2 Cayley Graphs 7.3 Growth Functions 7.4 Growth Types 7.5 The Growth Rate 7.6 Growth of Subgroups and Quotients 7.7 Groups of Linear Growth 7.8 The Growth of Nilpotent Groups and the Bass–Guivarc’h Formula 7.9 The Theorems of Milnor and Wolf 7.10 Notes 7.11 Exercises Chapter 8 Hyperbolic Plane Geometry and the Tits Alternative 8.1 Möbius Transformations 8.2 Hyperbolic (Plane) Geometry 8.3 The Lobachevsky–Poincaré Half-Plane 8.4 Isometries of the Lobachevsky–Poincaré Half-Plane 8.5 The Poincar´e Disc 8.6 Isometries of the Poincaré Disc 8.7 The Cayley Transform and the Definition of H 8.8 Classification of the Orientation-Preserving Isometries of H 8.9 Characterizations of Orientation-Preserving Isometries of H 8.10 The Tits Alternative for GL(2,R) 8.11 Growth of Finitely Generated Linear Groups 8.12 Notes Chapter 9 Topological Groups, Lie Groups, and Hilbert’s Fifth Problem 9.1 Topological Groups 9.2 Locally Compact Groups 9.3 The Haar Measure 9.4 Locally Compact Abelian Groups and Pontryagin Duality 9.5 Lie Groups 9.6 Hilbert’s Fifth Problem 9.7 Exercises Chapter 10 Dimension Theory 10.1 The Cantor Set 10.2 0-Dimensional Spaces 10.3 n-Dimensional Spaces 10.4 The Dimension of R^n 10.5 Dimension and Measure 10.6 Hausdorff Dimension 10.7 Notes 10.8 Exercises Chapter 11 Ultrafilters, Ultraproducts, Ultrapowers, and Asymptotic Cones 11.1 Filters 11.2 Ultrafilters 11.3 Free Ultrafilters 11.4 Limits along Filters in Metric Spaces 11.5 The Stone–Čech Compactification 11.6 The Completion of a Metric Space 11.7 Ultrapowers of Metric Spaces 11.8 Ultraproducts of Sequences of Pointed Metric Spaces 11.9 Ultraproducts of Groups 11.10 Ultrafields 11.11 Ultrapowers of General Linear Groups 11.12 Asymptotic Cones 11.13 Asymptotic Cones and Quasi-Isometries 11.14 Properties of Asymptotic Cones 11.15 Examples of Asymptotic Cones 11.16 Hyperbolic Metric Spaces 11.17 R-trees and Asymptotic Cones of Hyperbolic Metric Spaces 11.18 Notes 11.19 Exercises Chapter 12 Gromov’s Theorem 12.1 Asymptotic Cones of Groups of Sub-Polynomial Growth are Locally Compact 12.2 Finite Dimension of Asymptotic Cones of Groups of Sub-Polynomial Growth 12.3 Proof of Gromov’s Theorem 12.4 Notes 12.5 Exercises Part III Analytic and Probabilistic Theory Chapter 13 The Theorems of Polya and Varopoulos 13.1 The Simple Random Walk on Z^d: Setting the Problem 13.2 Markov Chains 13.3 Irreducible Markov Chains 13.4 Recurrent and Transient Markov Chains 13.5 Random Walks on Finitely Generated Groups 13.6 Recurrence of the Simple Random Walk on Z and Z^2 13.7 Transience of the Simple Random Walk on Z^3 13.8 Varopoulos’ Theorem and its Proof Strategy 13.9 Reversible Markov Chains and Networks 13.10 Criteria for Recurrence 13.11 Growth and Recurrence 13.12 Growth and Transience 13.13 The RandomWalk Alternative 13.14 Proof of Varopoulos’ Theorem 13.15 Notes 13.16 Exercises Chapter 14 Amenability, Isoperimetric Profiles, and Følner Functions 14.1 Amenability of Groups: Definitions and Examples 14.2 Stability Properties of Amenable Groups 14.3 Measures and Paradoxical Decompositions 14.4 Følner Nets and Følner Sequences 14.5 Kesten’s Amenability Criterion 14.6 Cogrowth and the Grigorchuk Criterion 14.7 The Ornstein–Weiss Lemma 14.8 Applications of the Ornstein–Weiss Lemma to Ergodic Theory and Dynamical Systems 14.9 The Tarski Number 14.10 Isoperimetric Profiles of Groups 14.11 Følner Functions 14.12 Følner Functions of Groups of Polynomial Growth 14.13 Notes Solutions or Hints to Selected Exercises References List of Symbols Subject Index Index of Authors