On invariant means and pre-syndetic subgroups

TL;DR

This paper explores various notions of amenability in topological groups beyond the locally compact setting, focusing on inheritance properties of skew-amenability and related concepts, and discusses open problems in the field.

Contribution

It introduces and analyzes skew-amenability, showing its inheritance by pre-syndetic subgroups, and compares different amenability notions including a stronger version related to invariant means.

Findings

Skew-amenability is inherited by pre-syndetic subgroups.

Cocompact subgroups of amenable groups with coinciding uniformities are amenable.

A stronger form of amenability involving invariant means is characterized by Reiter- and F{46}lner-type criteria.

Abstract

Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type properties of groups of paths and loops. It is shown that a version of amenability called skew-amenability is inherited by pre-syndetic subgroups in the sense of Basso and Zucker (in particular, by co-compact subgroups). It follows that co-compact subgroups of amenable topological groups whose left and right uniformities coincide are amenable. We discuss a version of amenability belonging to P. Malliavin and M.-P. Malliavin: the existence of a mean on bounded Borel functions that is invariant under the left action of a dense subgroup. We observe that this property is in general strictly stronger than amenability, and establish for it Reiter- and F{\o}lner-type criteria. Finally, there is a review of open problems.