Homogeneous actions on Urysohn spaces
Volume 167 / 2022
Colloquium Mathematicum 167 (2022), 21-61
MSC: Primary 03E15, 20E06; Secondary 20B22.
DOI: 10.4064/cm7706-1-2021
Published online: 26 March 2021
Abstract
We show that many countable groups acting on trees, including free products of infinitely 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 the last authors 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 infinitely countable groups arises as a dense subgroup of the isometry group of the rational Urysohn space.
