Classification of \(2\)-Quasi-Invariant Subsets

Let \(G\) be a group acting on a set \(\Omega\). A subset (finite or infinite) \(A \subseteq \Omega\) is called \(k\)-quasi-invariant, where \(k\) is a non-negative integer, if \(|A^g \backslash A| \leq k\) 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 \(\Omega\) 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 \(\Omega\) which have a symmetric difference of size at most \(2\) with some \(G\)-invariant subset) there are basically five explicitly determined possibilities.