Classification of $2$-Quasi-Invariant Subsets

Abstract

Let $G$ be a group acting on a set $\mathrm{\Omega }$. A subset (finite or infinite) $A\subseteq \mathrm{\Omega }$ is called $k$-quasi-invariant, where $k$ is a non-negative integer, if $|A^g\mathrm{\setminus }A|\le k$ for every $g\in G$. In previous work of the authors a bound was obtained, in terms of $k$, on the size of the symmetric difference between a $k$-quasi-invariant subset and the $G$-invariant subset of $\mathrm{\Omega }$ closest to it. However, apart from the cases $k=0,1$, this bound gave little information about the structure of a $k$-quasi-invariant subset. In this paper a classification of $2$-quasi-invariant subsets is given. Besides the generic examples (subsets of $\mathrm{\Omega }$ which have a symmetric difference of size at most $2$ with some $G$-invariant subset) there are basically five explicitly determined possibilities.