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.
Author(s): Tullio Ceccherini-Silberstein, Michele D'Adderio
Series: Springer Monographs in Mathematics
Publisher: Springer
Year: 2021
Language: English
Pages: 483
City: Cham
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
