# Simplicity of the automorphism groups of order and tournament expansions of homogeneous structures

## TL;DR

This paper introduces new concepts like free fusion and weakly stationary independence to prove that automorphism groups of certain expanded homogeneous structures are simple, including the Urysohn space, random graph, and random poset.

## Contribution

It develops a framework for establishing simplicity of automorphism groups of order and tournament expansions of homogeneous structures.

## Key findings

- Automorphism groups of the studied structures are simple.
- The framework applies to the Urysohn space, random graph, and random poset.
- New notions of free fusion and weakly stationary independence are introduced.

## Abstract

We define the notions of a free fusion of structures and a weakly stationary independence relation. We apply these notions to prove simplicity for the automorphism groups of order and tournament expansions of homogeneous structures like the bounded Urysohn space, the random graph, and the random poset.