Geometric Group Theory

Cornelia Drutu, Michael Kapovich (Autoren)

Buch | Softcover

819 Seiten

2018
American Mathematical Society (Verlag)
978-1-4704-8446-0 (ISBN)

Infos

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.

Autor

Cornelia Drutu, Mathematical Institute, Oxford, United Kingdom, and Michael Kapovich, University of California, Davis, CA

Inhalt

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
Gromov-hyperbolic spaces and groups
Lattices in Lie groups
Solvable groups
Geometric aspects of solvable groups
The Tits alternative
Gromov's theorem
The Banach-Tarski 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

Produkt-Details

• Erscheinungsdatum: 25.10.2025
• Reihe/Serie: Colloquium Publications
• Verlagsort: Providence
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• ISBN-10: 1-4704-8446-3 / 1470484463
• ISBN-13: 978-1-4704-8446-0 / 9781470484460
• Zustand: Neuware