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Topics in Geometric Group Theory

ISBN-10: 0226317218
ISBN-13: 9780226317212
Edition: 2000 (Reprint)
List price: $41.00 Buy it from $22.49
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Description: In this book, Pierre de la Harpe provides a concise and engaging introduction to geometric group theory, a new method for studying infinite groups via their intrinsic geometry that has played a major role in mathematics over the past two decades. A  More...

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Book details

List price: $41.00
Copyright year: 2000
Publisher: University of Chicago Press
Publication date: 9/15/2000
Binding: Paperback
Pages: 320
Size: 5.75" wide x 8.75" long x 0.75" tall
Weight: 0.946
Language: English

In this book, Pierre de la Harpe provides a concise and engaging introduction to geometric group theory, a new method for studying infinite groups via their intrinsic geometry that has played a major role in mathematics over the past two decades. A recognized expert in the field, de la Harpe adopts a hands-on approach, illustrating key concepts with numerous concrete examples. The first five chapters present basic combinatorial and geometric group theory in a unique and refreshing way, with an emphasis on finitely generated versus finitely presented groups. In the final three chapters, de la Harpe discusses new material on the growth of groups, including a detailed treatment of the "Grigorchuk group." Most sections are followed by exercises and a list of problems and complements, enhancing the book's value for students; problems range from slightly more difficult exercises to open research problems in the field. An extensive list of references directs readers to more advanced results as well as connections with other fields.

Introduction
Gauss' circle problem and Polya's random walks on lattices
The circle problem
Polya's recurrence theorem
Free products and free groups
Free Products of Groups
The Table-Tennis Lemma (Klein's criterion) and examples of free products
Finitely-generated groups
Finitely-generated and infinitely-generated groups
Uncountably many groups with two generators (B.H. Neumann's method)
On groups with two generators
On finite quotients of the modular group
Finitely-generated groups viewed as metric spaces
Word lengths and Cayley graphs
Quasi-isometries
Finitely-presented groups
Finitely-presented groups
The Poincare theorem on fundamental polygons
On fundamental groups and curvature in Riemannian geometry
Complement on Gromov's hyperbolic groups
Growth of finitely-generated groups
Growth functions and growth series of groups
Generalities on growth types
Exponential growth rate and entropy
Groups of exponential or polynomial growth
On groups of exponential growth
On uniformly exponential growth
On groups of polynomial growth
Complement on other kinds of growth
The first Grigorchuk group
Rooted d-ary trees and their automorphisms
The group [Gamma] as an answer to one of Burnside's problems
On some subgroups of [Gamma]
Congruence subgroups
Word problem and non-existence of finite presentations
Growth
Exercises and complements
References
Index of research problems
Subject index

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