# Simplicity of the automorphism groups of generalised metric spaces

## TL;DR

This paper demonstrates that certain stationary independence relations (SIRs) with specific properties can be used to prove the simplicity of automorphism groups of various countable structures, extending previous results on the Urysohn space.

## Contribution

It introduces a set of axioms for SIRs that, when satisfied, ensure the automorphism group of a structure is simple, broadening the scope of structures where simplicity can be established.

## Key findings

- SIRs with extra axioms imply automorphism group simplicity
- Applicability to homogeneous structures with metric-like amalgamation
- Extension of simplicity results beyond Urysohn spaces

## Abstract

Tent and Ziegler proved that the automorphism group of the Urysohn sphere is simple and that the automorphism group of the Urysohn space is simple modulo bounded automorphisms. A key component of their proof is the definition of a stationary independence relation (SIR). In this paper we prove that the existence of a SIR satisfying some extra axioms is enough to prove simplicity of the automorphism group of a countable structure. The extra axioms are chosen with applications in mind, namely homogeneous structures which admit a "metric-like amalgamation", for example all primitive 3-constrained metrically homogeneous graphs of finite diameter from Cherlin's list.