Some isometry groups of Urysohn space

TL;DR

This paper explores the structure of isometry groups of the universal Urysohn space, constructing various subgroups including abelian and free groups, revealing their properties and density within the full isometry group.

Contribution

It introduces new subgroups of the Urysohn space's isometry group, including dense free groups and transitive abelian groups, expanding understanding of its symmetry structure.

Findings

Constructed abelian groups acting transitively

Embedded free groups dense in the isometry group

Enhanced understanding of Urysohn space symmetries

Abstract

We constract various subgroups of the group of isometries of universal Urysohn spaces (unique complete separable metric space which is iniversal and homogeneous) including abelian groups which act transitively, and free groups which are dense in the full isometry group.