Coarse Geometry of Topological Groups

Christian Rosendal (University of Maryland, Baltimore)

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English

Cambridge University Press
16 December 2021

Set theory; Algebra; Geometry; Topology

Series: Cambridge Tracts in Mathematics

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

By: Christian Rosendal (University of Maryland Baltimore)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Edition: New edition
Dimensions: Height: 235mm, Width: 159mm, Spine: 25mm
Weight: 614g
ISBN: 9781108842471
ISBN 10: 110884247X
Series: Cambridge Tracts in Mathematics
Pages: 200
Publication Date: 16 December 2021
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

1. Introduction; 2. Coarse structure and metrisability; 3. Structure theory; 4. Sections, cocycles and group extensions; 5. Polish groups of bounded geometry; 6. Automorphism groups of countable structures; 7. Zappa-Szép products; Appendix A. Open problems.

Christian Rosendal is Professor of Mathematics at University of Illinois at Chicago. He received a Simons Fellowship in Mathematics in 2012 and is Fellow of the American Mathematical Society.

Reviews for Coarse Geometry of Topological Groups

'Great care is taken with exposition and proofs are spelled out in full detail. The book does not have a lot of prerequisites apart from some basic knowledge of topological groups, and it is accessible to graduate students or non-specialists interested in the subject. As it is a research monograph rather than a textbook, exercises are not included; however, it is certainly possible to teach parts of it in a topics graduate course on Polish groups or geometric group theory ... In view of the research presented in this monograph, now Polish groups can also be considered as geometric objects, and this new facet of the theory will undoubtedly lead to interactions with yet other branches of mathematics. The clear exposition and the numerous open questions that are discussed make the book an excellent entry point to research in the field.' Todor Tsankov, Bulletin of the American Mathematical Society