The geometrisation problem for topological groups

Abstract

This paper presents a framework for assigning intrinsic geometric structures to topological groups using only the data provided by their topological and algebraic structure. The geometrisation spits into small-scale and large-scale components, formalised respectively through local Lipschitz and quasimetric categories that, in turn, are definable from the canonical left uniform and left coarse structures of the group. For Polish groups, the paper characterises metrisability of the left coarse structure in terms of local boundedness, countable coverings by bounded sets, and the existence of compatible coarsely proper left-invariant metrics. It then introduces minimal metrics, which determine local Lipschitz structure, and maximal metrics, which determine quasimetric structure, and provides intrinsic characterisations of both. When both structures exist, they combine into a single canonical Lipschitz structure. Our framework is subsequently applied to specific examples such as homeomorphism groups, non-Archimedean Polish groups and automorphism groups of Fra\"iss\'e limits.