Asymmetric colouring of locally compact permutation groups

2022/08/18 by Lehner, Florian
#Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2208.08844

Abstract

Let G ≤ 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 ∈ G ∖\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.

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