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From Bulletin of the AMS

 Caption reads Figure 1. Forgetful functors between geometric categories. Image of a word map with Metric at the top, Lipschitz beneath it, then splitting to Local Lipschitz which has Uniform below it. The other split is to Quasimetric with Coarse below that.

Geometries of Topological Groups

by Christian Rosendal


The paper provides an overarching framework for the study of some of the intrinsic geometries that a topological group may carry. An initial analysis is based on geometric nonlinear functional analysis, that is, the study of Banach spaces as metric spaces up to various notions of isomorphism, such as bi-Lipschitz equivalence, uniform homeomorphism, and coarse equivalence. This motivates the introduction of the various geometric categories applicable to all topological groups, namely, their uniform and coarse structure, along with those applicable to a more select class, that is, (local) Lipschitz and quasimetric structure. Our study touches on Lie theory, geometric group theory, and geometric nonlinear functional analysis and makes evident that these can all be seen as instances of a single coherent theory.

The aim of the present paper is to organise and put into a coherent form a number of old and new results, ideas and research programmes regarding topological groups and their linear counterparts, namely Banach spaces.

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