Homogeneous actions on Urysohn spaces

TL;DR

This paper demonstrates that various countable groups acting on trees can be embedded densely into isometry groups of Urysohn spaces, extending prior work on automorphism groups of graphs.

Contribution

It establishes new embeddings of groups into isometry groups of both bounded and unbounded Urysohn spaces, broadening the understanding of group actions on these metric spaces.

Findings

Countable groups acting on trees can be densely embedded into isometry groups of bounded Urysohn spaces.

Free products of infinite countable groups can be embedded into isometry groups of the rational Urysohn space.

Results extend previous work on automorphism groups of the random graph.

Abstract

We show that many countable groups acting on trees, including free products of infinite countable groups and surface groups, are isomorphic to dense subgroups of isometry groups of bounded Urysohn spaces. This extends previous results of the first and last author with Y. Stalder on dense subgroups of the automorphism group of the random graph. In the unbounded case, we also show that every free product of infinite countable groups arises as a dense subgroup of the isometry group of the rational Urysohn space.