On the isometry group of the Urysohn space

TL;DR

This paper establishes a criterion for the simplicity of automorphism groups of certain structures and applies it to show that the isometry group of the Urysohn space, modulo bounded isometries, is simple.

Contribution

It introduces a general criterion for bounded simplicity and demonstrates its application to the isometry group of the Urysohn space.

Findings

The isometry group modulo bounded isometries is simple.

A new criterion for bounded simplicity of automorphism groups.

Application to the Urysohn space's automorphism group.

Abstract

We give a general criterion for the (bounded) simplicity of the automorphism groups of certain countable structures and apply it to show that the isometry group of the Urysohn space modulo the normal subgroup of bounded isometries is a simple group.