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

Thomas Bernert, Rob Sullivan, Jeroen Winkel, Shujie Yang

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 $\frac{2π}{n}$-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.

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

Abstract

We show the simplicity of the automorphism groups of the generic -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 -local order for (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.

Paper Structure

This paper contains 22 sections, 53 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. New results: n-hypertournaments, a new proof for 2pi/n-local orders
  4. New results: oriented graphs, including the semigeneric tournament
  5. Why these particular structures? A new approach: SWIR expansions.
  6. The structure of the paper and an outline of the various examples
  7. Stationary weak independence relations (SWIRs) and automorphisms moving almost R-/L-maximally
  8. Free SWIRs and the pointwise-stabiliser lemma
  9. Free SWIRs
  10. The pointwise-stabiliser lemma
  11. The generic n-hypertournament
  12. The dense 2pi/n-local order Sn
  13. Ultrahomogeneous oriented graphs
  14. The oriented graph S(2)-double dot
  15. The generic n-partite tournament
  16. ...and 7 more sections

Key Result

Theorem A

Let $n \geq 3$. The automorphism group of the generic $n$-hypertournament $\mathbb{T}_n$ is simple.

Figures (2)

  • Figure 1: An example of a $3$-hypertournament, usually denoted by $H_4$ (see CHKN21). The curved arrows denote anticlockwise/clockwise orientations of each face (the grey arrows are for the back faces): the anticlockwise edges are $(v_0, v_1, v_3), (v_1, v_2, v_3), (v_2, v_0, v_3), (v_0, v_2, v_1)$ and their cyclic permutations.
  • Figure 2: Examples of dense $\frac{2\pi}{n}$-local orders.

Theorems & Definitions (127)

  • Definition 1.5
  • Definition 1.6
  • Theorem A
  • Theorem C
  • Theorem D
  • Definition 2.2: Li21
  • Definition 2.3
  • Example 2.4
  • Lemma 2.5
  • Lemma 2.7
  • ...and 117 more