With an appendix by Bogdan Nica
Hardcover ISBN:  9781470411046 
Product Code:  COLL/63 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470441647 
Product Code:  COLL/63.E 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Hardcover ISBN:  9781470411046 
eBook: ISBN:  9781470441647 
Product Code:  COLL/63.B 
List Price:  $188.00 $143.50 
MAA Member Price:  $169.20 $129.15 
AMS Member Price:  $150.40 $114.80 
With an appendix by Bogdan Nica
Hardcover ISBN:  9781470411046 
Product Code:  COLL/63 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470441647 
Product Code:  COLL/63.E 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Hardcover ISBN:  9781470411046 
eBook ISBN:  9781470441647 
Product Code:  COLL/63.B 
List Price:  $188.00 $143.50 
MAA Member Price:  $169.20 $129.15 
AMS Member Price:  $150.40 $114.80 

Book DetailsColloquium PublicationsVolume: 63; 2018; 819 ppMSC: Primary 20; 57
The key idea in geometric group theory is to study infinite groups by endowing them with a metric and treating them as geometric spaces. This applies to many groups naturally appearing in topology, geometry, and algebra, such as fundamental groups of manifolds, groups of matrices with integer coefficients, etc. The primary focus of this book is to cover the foundations of geometric group theory, including coarse topology, ultralimits and asymptotic cones, hyperbolic groups, isoperimetric inequalities, growth of groups, amenability, Kazhdan's Property (T) and the Haagerup property, as well as their characterizations in terms of group actions on median spaces and spaces with walls.
The book contains proofs of several fundamental results of geometric group theory, such as Gromov's theorem on groups of polynomial growth, Tits's alternative, Stallings's theorem on ends of groups, Dunwoody's accessibility theorem, the Mostow Rigidity Theorem, and quasiisometric rigidity theorems of Tukia and Schwartz. This is the first book in which geometric group theory is presented in a form accessible to advanced graduate students and young research mathematicians. It fills a big gap in the literature and will be used by researchers in geometric group theory and its applications.
ReadershipGraduate students and researchers interested in geometric group theory.

Table of Contents

Chapters

Geometry and topology

Metric spaces

Differential geometry

Hyperbolic space

Groups and their actions

Median spaces and spaces with measured walls

Finitely generated and finitely presented groups

Coarse geometry

Coarse topology

Ultralimits of metric spaces

Gromovhyperbolic spaces and groups

Lattices in Lie groups

Solvable groups

Geometric aspects of solvable groups

The Tits alternative

Gromov’s theorem

The BanachTarski paradox

Amenability and paradoxical decomposition

Ultralimits, fixed point properties, proper actions

Stallings’s theorem and accessibility

Proof of Stallings’s theorem using harmonic functions

Quasiconformal mappings

Groups quasiisometric to $\mathbb {H}^n$

Quasiisometries of nonuniform lattices in $\mathbb {H}^n$

A survey of quasiisometric rigidity

Appendix: Three theorems on linear groups


Additional Material

Reviews

[This book] offers a comprehensive account of major developments in geometric group theory in the 20th century. It is inevitable that some topics are mentioned only briefly, which is compensated by the extensive bibliography of over 600 references, both old and recent.
Igor Belegradek, Mathematical Reviews 
It will undoubtedly be an essential reference for those working in GGT and related areas in differential geometry and topology. Its wealth of immensely valuable historical commentary is particularly appreciated.
Scott Taylor, MAA Reviews


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The key idea in geometric group theory is to study infinite groups by endowing them with a metric and treating them as geometric spaces. This applies to many groups naturally appearing in topology, geometry, and algebra, such as fundamental groups of manifolds, groups of matrices with integer coefficients, etc. The primary focus of this book is to cover the foundations of geometric group theory, including coarse topology, ultralimits and asymptotic cones, hyperbolic groups, isoperimetric inequalities, growth of groups, amenability, Kazhdan's Property (T) and the Haagerup property, as well as their characterizations in terms of group actions on median spaces and spaces with walls.
The book contains proofs of several fundamental results of geometric group theory, such as Gromov's theorem on groups of polynomial growth, Tits's alternative, Stallings's theorem on ends of groups, Dunwoody's accessibility theorem, the Mostow Rigidity Theorem, and quasiisometric rigidity theorems of Tukia and Schwartz. This is the first book in which geometric group theory is presented in a form accessible to advanced graduate students and young research mathematicians. It fills a big gap in the literature and will be used by researchers in geometric group theory and its applications.
Graduate students and researchers interested in geometric group theory.

Chapters

Geometry and topology

Metric spaces

Differential geometry

Hyperbolic space

Groups and their actions

Median spaces and spaces with measured walls

Finitely generated and finitely presented groups

Coarse geometry

Coarse topology

Ultralimits of metric spaces

Gromovhyperbolic spaces and groups

Lattices in Lie groups

Solvable groups

Geometric aspects of solvable groups

The Tits alternative

Gromov’s theorem

The BanachTarski paradox

Amenability and paradoxical decomposition

Ultralimits, fixed point properties, proper actions

Stallings’s theorem and accessibility

Proof of Stallings’s theorem using harmonic functions

Quasiconformal mappings

Groups quasiisometric to $\mathbb {H}^n$

Quasiisometries of nonuniform lattices in $\mathbb {H}^n$

A survey of quasiisometric rigidity

Appendix: Three theorems on linear groups

[This book] offers a comprehensive account of major developments in geometric group theory in the 20th century. It is inevitable that some topics are mentioned only briefly, which is compensated by the extensive bibliography of over 600 references, both old and recent.
Igor Belegradek, Mathematical Reviews 
It will undoubtedly be an essential reference for those working in GGT and related areas in differential geometry and topology. Its wealth of immensely valuable historical commentary is particularly appreciated.
Scott Taylor, MAA Reviews