Homogeneous actions on Urysohn spaces
Résumé
We show that many countable groups acting on trees, including free prod-ucts 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.
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