Finitely approximable groups and actions Part II: Generic representations

TL;DR

This paper investigates the space of all isometric actions of finitely generated groups on the rational Urysohn space, showing that for finitely generated Abelian groups, a generic conjugacy class of actions exists.

Contribution

It establishes the existence of a generic conjugacy class of actions for finitely generated Abelian groups on the rational Urysohn space.

Findings

Existence of a comeagre conjugacy class for Abelian groups

Complexity of the action space varies with group type

Special case: free groups have a simpler structure

Abstract

Given a finitely generated group $\Gamma$, we study the space ${\rm Isom}(\Gamma,{\mathbb Q\mathbb U})$ of all actions of $\Gamma$ by isometries of the rational Urysohn metric space ${\mathbb Q\mathbb U}$, where ${\rm Isom}(\Gamma,{\mathbb Q\mathbb U})$ is equipped with the topology it inherits seen as a closed subset of ${\rm Isom}({\mathbb Q\mathbb U})^\Gamma$. When $\Gamma$ is the free group $\F_n$ on $n$ generators this space is just ${\rm Isom}({\mathbb Q\mathbb U})^n$, but is in general significantly more complicated. We prove that when $\Gamma$ is finitely generated Abelian there is a generic point in ${\rm Isom}(\Gamma,{\mathbb Q\mathbb U})$, i.e., there is a comeagre set of mutually conjugate isometric actions of $\Gamma$ on ${\mathbb Q\mathbb U}$.