Topics in Geometric Group Theory

Introduction

I. Gauss' circle problem and Pólya's random walks on lattices
I.A. The circle problem
I.B. Pólya's recurrence theorem

II. Free products and free groups
II.A. Free products of groups
II.B. The Table-Tennis Lemma (Klein's criterion) and examples of free products

III. Finitely-generated groups
III.A. Finitely-generated and infinitely-generated groups
III.B. Uncountably many groups with two generators (B.H. Neumann's method)
III.C. On groups with two generators
III.D. On finite quotients of the modular group

IV. Finitely-generated groups viewed as metric spaces
IV.A. Word lengths and Cayley graphs
IV.B. Quasi-isometries

V. Finitely-presented groups
V.A. Finitely-presented groups
V.B. The Poincaré theorem on fundamental polygons
V.C. On fundamental groups and curvature in Riemannian geometry
V.D. Complement on Gromov's hyperbolic groups

VI. Growth of finitely-generated groups
VI.A. Growth functions and growth series of groups
VI.B. Generalities on growth types
VI.C. Exponential growth rate and entropy

VII. Groups of exponential or polynomial growth
VII.A. On groups of exponential growth
VII.B. On uniformly exponential growth
VII.C. On groups of polynomial growth
VII.D. Complement on other kinds of growth

VIII. The first Grigorchuk group
VIII.A. Rooted d-ary trees and their automorphisms
VIII.B. The group r as an answer to one of Burnside's problems
VIII.C. On some subgroups of r
VIII.D. Congruence subgroups
VIII.E. Word problem and non-existence of finite presentations
VIII.F. Growth
VIII.G. Exercises and complements

References
Index of research problems
Subject index