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Topics in Geometric Group Theory : Chicago Lectures in Mathematics - Pierre de la Harpe

Topics in Geometric Group Theory

Chicago Lectures in Mathematics

Paperback Published: 1st January 2000
ISBN: 9780226317212
Number Of Pages: 320

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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.

Introductionp. 1
Gauss' circle problem and Polya's random walks on latticesp. 5
The circle problemp. 5
Polya's recurrence theoremp. 7
Free products and free groupsp. 17
Free Products of Groupsp. 17
The Table-Tennis Lemma (Klein's criterion) and examples of free productsp. 25
Finitely-generated groupsp. 43
Finitely-generated and infinitely-generated groupsp. 43
Uncountably many groups with two generators (B.H. Neumann's method)p. 60
On groups with two generatorsp. 68
On finite quotients of the modular groupp. 71
Finitely-generated groups viewed as metric spacesp. 75
Word lengths and Cayley graphsp. 75
Quasi-isometriesp. 84
Finitely-presented groupsp. 117
Finitely-presented groupsp. 117
The Poincare theorem on fundamental polygonsp. 135
On fundamental groups and curvature in Riemannian geometryp. 145
Complement on Gromov's hyperbolic groupsp. 148
Growth of finitely-generated groupsp. 151
Growth functions and growth series of groupsp. 151
Generalities on growth typesp. 167
Exponential growth rate and entropyp. 180
Groups of exponential or polynomial growthp. 187
On groups of exponential growthp. 187
On uniformly exponential growthp. 194
On groups of polynomial growthp. 197
Complement on other kinds of growthp. 206
The first Grigorchuk groupp. 211
Rooted d-ary trees and their automorphismsp. 211
The group [Gamma] as an answer to one of Burnside's problemsp. 217
On some subgroups of [Gamma]p. 225
Congruence subgroupsp. 236
Word problem and non-existence of finite presentationsp. 240
Growthp. 248
Exercises and complementsp. 259
Referencesp. 265
Index of research problemsp. 295
Subject indexp. 299
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780226317212
ISBN-10: 0226317218
Series: Chicago Lectures in Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 320
Published: 1st January 2000
Publisher: The University of Chicago Press
Country of Publication: US
Dimensions (cm): 22.8 x 15.5  x 1.91
Weight (kg): 0.41
Edition Number: 2

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