Fixed point-free isometric actions of topological groups on Banach spaces

TL;DR

This paper characterizes precompact topological groups by their fixed point properties for affine isometric actions on Banach spaces, showing non-precompact groups always admit fixed point-free actions, with specific results for Polish groups.

Contribution

It establishes a link between precompactness of topological groups and fixed point properties for affine isometric actions on Banach spaces, introducing the Holmes space as a key example.

Findings

Non-precompact groups admit fixed point-free affine isometric actions.

Precompact groups are characterized by fixed point properties.

Certain Polish groups do not admit proper affine isometric actions.

Abstract

We show that every non-precompact topological group admits a fixed point-free continuous action by affine isometries on a suitable Banach space. Thus, precompact groups are defined by the fixed point property for affine isometric actions on Banach spaces. For separable topological groups, in the above statements it is enough to consider affine actions on one particular Banach space: the unique Banach space envelope of the universal Urysohn metric space, known as the Holmes space. At the same time, we show that Polish groups need not admit topologically proper (in particular, free) affine isometric actions on Banach spaces (nor even on complete metric spaces): this is the case for the unitary group of the separable infinite dimensional Hilbert space with strong operator topology, the infinite symmetric group, etc.