Schur ultrafilters and Bohr compactifications of topological groups

TL;DR

This paper explores Schur ultrafilters within the Stone-ech compactification framework to provide a novel characterization of Bohr compactifications of topological groups, linking ultrafilter convergence to group topology.

Contribution

It introduces a new description of Bohr compactifications using Schur ultrafilters and characterizes topological groups via ultrafilter convergence properties.

Findings

A new characterization of Bohr compactifications.

A criterion for when a chart group is a topological group.

Ultrafilter convergence to the identity characterizes topological groups.

Abstract

In this paper we investigate Schur ultrafilters on groups. Using the algebraic structure of Stone-\v{C}ech compactifications of discrete groups and Schur ultrafilters, we give a new description of Bohr compactifications of topological groups. This approach allows us to characterize chart groups that are topological groups. Namely, a chart group $G$ is a topological group if and only if each Schur ultrafilter on $G$ converges to the unit of $G$.