Topics in Geometric Group Theory
310 pages

6 x 9

© 2000
Contents
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 TableTennis Lemma (Klein's criterion) and examples of free products
III. Finitelygenerated groups
III.A. Finitelygenerated and infinitelygenerated 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. Finitelygenerated groups viewed as metric spaces
IV.A. Word lengths and Cayley graphs
IV.B. Quasiisometries
V. Finitelypresented groups
V.A. Finitelypresented 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 finitelygenerated 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 dary 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 nonexistence of finite presentations
VIII.F. Growth
VIII.G. Exercises and complements
References
Index of research problems
Subject index
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 TableTennis Lemma (Klein's criterion) and examples of free products
III. Finitelygenerated groups
III.A. Finitelygenerated and infinitelygenerated 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. Finitelygenerated groups viewed as metric spaces
IV.A. Word lengths and Cayley graphs
IV.B. Quasiisometries
V. Finitelypresented groups
V.A. Finitelypresented 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 finitelygenerated 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 dary 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 nonexistence of finite presentations
VIII.F. Growth
VIII.G. Exercises and complements
References
Index of research problems
Subject index
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