Coarse Geometry of Topological Groups

Christian Rosendal

doi:10.1017/9781108903547

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.

‘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 Source: Bulletin of the American Mathematical Society

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Coarse Geometry of Topological Groups

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-viii

Acknowledgements
pp ix-x

1 - Introduction
pp 1-18

2 - Coarse Structure and Metrisability
pp 19-67

3 - Structure Theory
pp 68-124

4 - Sections, Cocycles and Group Extensions
pp 125-165

5 - Polish Groups of Bounded Geometry
pp 166-243

6 - Automorphism Groups of Countable Structures
pp 244-271

7 - Zappa–Szép Products
pp 272-284

Appendix Open Problems
pp 285-288

References
pp 289-294

Index
pp 295-298

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