Locally Roelcke precompact Polish groups

TL;DR

This paper characterizes locally Roelcke precompact Polish groups through their geometric and uniform structures, extending the understanding of their properties and the conditions for their completions to be locally compact.

Contribution

It provides a new characterization of these groups via geometric and uniform structures, and explores conditions for their completions to be locally compact.

Findings

Locally Roelcke precompact groups are characterized by their coarsely bounded sets being Roelcke precompact.

Their completions in the Roelcke uniformity are locally compact.

Conditions are identified under which these groups form semi-topological semigroups.

Abstract

A Polish group is said to be locally Roelcke precompact if there is a neighborhood of the identity element that is totally bounded in the Roelcke (or lower) group uniformity. These form a subclass of the locally bounded groups, while generalizing the Roelcke precompact and locally compact Polish groups. We characterize these groups in terms of their geometric structure as those locally bounded groups whose coarsely bounded sets are all Roelcke precompact, and in terms of their uniform structure as those groups whose completions in the Roelcke uniformity are locally compact. We also assess the conditions under which this locally compact space carries the structure of a semi-topological semigroup.