Determining the normal subgroups of the automorphism groups of ultrahomogeneous structures via stabilisers

TL;DR

This paper investigates the automorphism groups of ultrahomogeneous structures, proving simplicity for certain cases and determining normal subgroups for others, using a novel approach involving expansions with stationary weak independence relations.

Contribution

It introduces a new method applying stationary weak independence relations to expansions of structures, enabling analysis of automorphism groups where previous techniques failed.

Findings

Proves simplicity of automorphism groups of generic n-hypertournaments and semigeneric tournaments.

Determines normal subgroups of automorphism groups of various ultrahomogeneous oriented graphs.

Provides a new proof of the automorphism group's simplicity for dense 2π/n-local orders.

Abstract

We show the simplicity of the automorphism groups of the generic $n$-hypertournament and the semigeneric tournament, and determine the normal subgroups of the automorphism groups of several other ultrahomogeneous oriented graphs. We also give a new proof of the simplicity of the automorphism group of the dense $\frac{2\pi}{n}$-local order $\mathbb{S}(n)$ for $n \geq 2$ (a result due to Droste, Giraudet and Macpherson). Previous techniques of Li, Macpherson, Tent and Ziegler involving stationary weak independence relations (SWIRs) cannot be applied directly to these structures; our approach involves applying these techniques to a certain expansion of each structure, where the expansion has a SWIR and its automorphism group is isomorphic to a stabiliser subgroup of the automorphism group of the original structure.