Coarse fixed point properties

TL;DR

This paper explores fixed point properties for topological groups acting on metric spaces, establishing equivalences between coarse and continuous actions and extending classical results to broader contexts.

Contribution

It introduces a coarse fixed point property framework for topological groups, generalizes Gromov's results, and characterizes geometric property (T) via coarse fixed point properties.

Findings

Coarse fixed point properties are equivalent to classical ones for locally compact groups.

Generalizations of Gromov's results on fixed point properties are established.

Characterization of geometric property (T) for finite Cayley graph sequences.

Abstract

We investigate fixed point properties for isometric actions of topological groups on a wide class of metric spaces, with a particular emphasis on Hilbert spaces. Instead of requiring the action to be continuous, we assume that it is ``controlled", i.e. compatible with respect to some natural left-invariant coarse structure. For locally compact groups, we prove that these coarse fixed point properties are equivalent to the usual ones, defined for continuous actions. We deduce generalisations of two results of Gromov originally stated for discrete groups. For Polish groups with bounded geometry (in the sense of Rosendal), we prove a version of Serre's theorem on the stability of coarse property FH under central extensions. As an application we prove that the group $\text{Homeo}^+_{\mathbb Z}(\mathbb R)$ has property FH. Finally, we characterise geometric property (T) for sequences of finite Cayley graphs in terms of coarse property FH of a certain group.