On topological groups with an approximate fixed point property

TL;DR

This paper investigates the approximate fixed point property in topological groups and explores its connection with amenability, providing insights into the behavior of affine actions on convex sets.

Contribution

It introduces and analyzes the AFP property in topological groups, establishing its relationship with amenability and expanding understanding of group actions in topological vector spaces.

Findings

AFP property relates closely to amenability in topological groups

Characterization of affine actions with approximate fixed points

Insights into the structure of groups with AFP property

Abstract

A topological group $G$ has the Approximate Fixed Point (AFP) property on a bounded convex subset $C$ of a locally convex space if every continuous affine action of $G$ on $C$ admits a net $(x_i)$, $x_i\in C$, such that $x_{i}-gx_{i}\longrightarrow 0$ for all $g\in G$. We study the relationship of this property with amenability.