Coarse Geometry of Topological Groups

Provides a general framework for doing geometric group theory for non-locally-compact topological groups arising in mathematical practice.

Christian Rosendal (Author)

9781108842471, Cambridge University Press

Hardback, published 16 December 2021

200 pages
23.5 x 15.9 x 2.5 cm, 0.614 kg

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.

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.

Subject Areas: Topology [PBP], Geometry [PBM], Algebra [PBF], Set theory [PBCH]