Coarse Geometry of Topological Groups
Cambridge Tracts in Mathematics Series

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.

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.