Synopses & Reviews
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.
Synopsis
Introduction I. Gauss' circle problem and Polya's random walks on latticesI.A. The circle problemI.B. Polya's recurrence theoremII. Free products and free groupsII.A. Free products of groupsII.B. The Table-Tennis Lemma (Klein's criterion) and examples of free productsIII. Finitely-generated groupsIII.A. Finitely-generated and infinitely-generated groupsIII.B. Uncountably many groups with two generators (B.H. Neumann's method)III.C. On groups with two generatorsIII.D. On finite quotients of the modular groupIV. Finitely-generated groups viewed as metric spacesIV.A. Word lengths and Cayley graphsIV.B. Quasi-isometriesV. Finitely-presented groupsV.A. Finitely-presented groupsV.B. The Poincare theorem on fundamental polygonsV.C. On fundamental groups and curvature in Riemannian geometryV.D. Complement on Gromov's hyperbolic groupsVI. Growth of finitely-generated groupsVI.A. Growth functions and growth series of groupsVI.B. Generalities on growth typesVI.C. Exponential growth rate and entropyVII. Groups of exponential or polynomial growthVII.A. On groups of exponential growthVII.B. On uniformly exponential growthVII.C. On groups of polynomial growthVII.D. Complement on other kinds of growthVIII. The first Grigorchuk groupVIII.A. Rooted d-ary trees and their automorphismsVIII.B. The group r as an answer to one of Burnside's problemsVIII.C. On some subgroups of rVIII.D. Congruence subgroupsVIII.E. Word problem and non-existence of finite presentationsVIII.F. GrowthVIII.G. Exercises and complementsReferences Index of research problems Subject index
Synopsis
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.
Synopsis
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.
About the Author
Pierre de la Harpe is a professor of mathematics at the Université de Genève, Switzerland. He is the author, coauthor, or coeditor of several books, including La propriété (T) de Kazhdan pour les groupes localement compacts and Sur les groupes hyperboliques d'après Mikhael Gromov.
Table of 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 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