Asymmetric colouring of locally compact permutation groups

TL;DR

This paper proves that every locally compact, closed permutation group with infinite motion can be coloured with two colours asymmetrically, extending previous results and confirming a longstanding conjecture.

Contribution

It generalizes recent findings by showing all such groups admit an asymmetric 2-colouring, confirming a conjecture from 2015.

Findings

Every locally compact, closed permutation group with infinite motion admits an asymmetric 2-colouring.

The result extends previous work by Babai and confirms a conjecture from 2015.

The paper broadens understanding of symmetry-breaking in permutation groups.

Abstract

Let $G \leq \mathrm{Sym} (X)$ for a countable set $X$. Call a colouring of $X$ asymmetric, if the identity is the only element of $G$ which preserves all colours. The motion (also called minimal degree) of $G$ is the minimal number of elements moved by an element $g \in G \setminus\{\mathrm{id}\}$. We show that every locally compact, closed permutation group with infinite motion admits an asymmetric $2$-colouring. This generalises a recent result by Babai and confirms a conjecture by Imrich, Smith, Tucker, and Watkins from 2015.