Lipschitz structure and minimal metrics on topological groups

TL;DR

This paper investigates when metrisable topological groups possess a natural Lipschitz geometry, introduces minimal metrics characterized by linear growth, and links these concepts to large-scale geometry and Hilbert's fifth problem.

Contribution

It provides an intrinsic characterization of minimal metrics on topological groups and identifies conditions for canonical global Lipschitz geometries.

Findings

Characterization of minimal metrics via linear growth conditions

Identification of groups with canonical global Lipschitz geometry

Connection between minimal metrics and Hilbert's fifth problem

Abstract

We discuss the problem of deciding when a metrisable topological group $G$ has a canonically defined local Lipschitz geometry. This naturally leads to the concept of minimal metrics on $G$, that we characterise intrinsically in terms of a linear growth condition on powers of group elements. Combining this with work on the large scale geometry of topological groups, we also identify the class of metrisable groups admitting a canonical global Lipschitz geometry. In turn, minimal metrics connect with Hilbert's fifth problem for completely metrisable groups and we show, assuming that the set of squares is sufficiently rich, that every element of some identity neighbourhood belongs to a $1$-parameter subgroup.