Coarse structures on groups

TL;DR

This paper introduces a new coarse structure on topological groups, develops asymptotic dimension theory for it, and applies it to free topological groups, revealing their asymptotic dimension as 1.

Contribution

It presents a novel group-compact coarse structure for topological groups and extends asymptotic dimension theory to this setting, including new results for free topological groups.

Findings

Asymptotic dimension of free topological groups is 1.

Developed a generalized asymptotic dimension theory for the new coarse structure.

Connected coarse geometry with topological group properties.

Abstract

We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the group-compact coarse structure generalizing several familiar results for discrete groups. We show that the asymptotic dimension in our sense of the free topological group on a non-empty topological space that is homeomorphic to a closed subspace of a Cartesian product of metrizable spaces is 1.