Universal actions and representations of locally finite groups on metric spaces

TL;DR

This paper constructs a universal isometric action of a countable locally finite group on a separable metric space, encompassing all such actions and demonstrating the genericity of this universal action.

Contribution

It introduces a universal action for countable locally finite groups on metric spaces, using amalgamation of actions, and shows its genericity, a result not extendable to non-locally finite groups.

Findings

Universal action contains all countable locally finite group actions

The universal action is generic in its class

Results do not extend to non-locally finite groups

Abstract

We construct a universal action of a countable locally finite group (the Hall's group) on a separable metric space by isometries. This single action contains all actions of all countable locally finite groups on all separable metric spaces as subactions. The main ingredient is the amalgamation of actions by isometries. We show that an equivalence class of this universal action is generic. We show that the restriction to locally finite groups in our results is necessary as analogous results do not hold for infinite non-locally finite groups. We discuss the problem also for actions by linear isometries on Banach spaces.