[Paper Review] Homogeneous actions on Urysohn spaces

Pierre Fima, François Le Maı̂tre+2 more|arXiv (Cornell University)|May 7, 2018

Geometric and Algebraic Topology20 references3 citations

TL;DR

This paper establishes that many countable groups acting on trees—such as free products of infinite countable groups and surface groups—are isomorphic to dense subgroups of isometry groups of bounded Urysohn spaces. Using Katětov extensions and equivariant amalgamation techniques, it proves that such actions are homogeneous, extending earlier results on highly transitive groups and the random graph to the broader context of Urysohn metric spaces.

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.

Motivation & Objective

• To characterize countable groups that admit homogeneous isometric actions on Urysohn spaces.
• To extend previous results on highly transitive groups and the random graph to the setting of bounded and unbounded Urysohn spaces.
• To establish that free products of infinite countable groups and surface groups arise as dense subgroups of isometry groups of bounded Urysohn spaces.
• To analyze the structure of point stabilizers and Schlichting completions for homogeneous actions on Urysohn spaces.
• To prove that the group of finitely supported permutations embeds densely into the isometry group of the rational Urysohn space, with uncountable conjugacy classes in the completion.

Proposed method

• Utilizes Katětov extension techniques to construct isometric extensions of partial isometries in S-Urysohn spaces.
• Applies equivariant amalgamation to extend finite partial isometries to global group actions, ensuring homogeneity.
• Employs the Baire category theorem to show that generic extensions yield dense subgroups in the isometry group.
• Uses Schlichting completion to relate point stabilizers in homogeneous actions to topological properties of the isometry group.
• Applies strong disconnection and free product constructions in the unbounded case to realize dense embeddings of free products.
• Analyzes conjugacy classes in the isometry group to rule out nontrivial countable normal subgroups, leveraging the cardinality of distance sets in S.

Experimental results

Research questions

• RQ1 Which countable groups can act homogeneously on a Urysohn space via a dense isometric action?
• RQ2 Under what conditions does a free product of infinite countable groups embed densely into the isometry group of a bounded Urysohn space?
• RQ3 What is the structure of the Schlichting completion of a group acting homogeneously on a Urysohn space?
• RQ4 Can the group of finitely supported permutations be realized as a dense subgroup of the isometry group of the rational Urysohn space?
• RQ5 What topological constraints (e.g., conjugacy class size) arise in the isometry group of a Urysohn space when the distance set S has size at least 3?

Key findings

• Every free product of infinite countable groups embeds densely into the isometry group of the bounded Urysohn space when the distance set S is countable and contains 0.
• Surface groups and free products of infinite countable groups are isomorphic to dense subgroups of the isometry group of a bounded Urysohn space.
• The group of finitely supported permutations embeds densely into the isometry group of the rational Urysohn space, and its Schlichting completion is the closure of this group in the full isometry group.
• For |S| ≥ 3, the isometry group of the S-Urysohn space has uncountable conjugacy classes, implying no nontrivial countable normal subgroups.
• The natural action of the finitely supported permutation group on ℕ is the only 2-transitive action up to conjugacy, and it is not highly faithful.
• The Schlichting completion of the finitely supported permutation group with respect to any maximal proper subgroup of infinite index is the closure of the group in the isometry group of the Urysohn space.