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Topics in geometric group theory

Author: Pierre de La Harpe
Publisher: Chicago : University of Chicago Press, ©2000.
Series: Chicago lectures in mathematics.
Edition/Format:   Book : EnglishView all editions and formats
Database:WorldCat
Summary:
"Groups as abstract structures were first recognized by mathematicians in the nineteenth century. Groups are, of course, sets given with appropriate "multiplications," and they are often given together with actions on interesting geometric objects. But groups are also interesting geometric objects by themselves. More precisely, a finitely-generated group can be seen as a metric space, the distance between two points  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Pierre de La Harpe
ISBN: 0226317196 9780226317199 0226317218 9780226317212
OCLC Number: 43227307
Description: vi, 310 p. : ill. ; 24 cm.
Contents: 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.
Series Title: Chicago lectures in mathematics.
Responsibility: Pierre de la Harpe.
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Abstract:

This work seeks to offer a concise introduction to geometric group theory - a method for studying infinite groups via their intrinsic geometry. Basic combinatorial and geometric group theory is  Read more...

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schema:reviewBody""Groups as abstract structures were first recognized by mathematicians in the nineteenth century. Groups are, of course, sets given with appropriate "multiplications," and they are often given together with actions on interesting geometric objects. But groups are also interesting geometric objects by themselves. More precisely, a finitely-generated group can be seen as a metric space, the distance between two points being defined "up to quasi-isometry" by some "word length," and this gives rise to a very fruitful approach to group theory." "In this book, Pierre de la Harpe provides a concise and engaging introduction to this approach, 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 uses a hands-on presentation style, illustrating key concepts of geometric group theory with numerous concrete examples." "The first five chapters present basic combinatorial and geometric group theory in a unique 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," an infinite finitely-generated torsion group of intermediate growth which is becoming more and more important in group theory. 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 questions in the field. An extensive list of references directs readers to more advanced results as well as connections with other subjects. Book jacket."--BOOK JACKET."
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