Local statistics of random permutations from free products (Q6623548)

Let \(\Gamma\) be a finitely generated group. The set \(\mathrm{Hom}(\Gamma, S_{N})\) of group homomorphisms from \(\Gamma\) to the symmetric group \(S_{N}\) is finite and it is a natural object of study since it is the set of all permutation-representations of \(\Gamma\) on a set of size \(N\). Such a set, as \(N\) varies, encodes many properties of \(\Gamma\), such as residual properties, profinite topology and subgroup growth.\N\NAssume that \(\Gamma=G_{1} \ast G_{2} \ast \dots\ast G_{k}\) is a free product of groups where each of \(G_{1}, G_{2}, \dots, G_{k}\) is either finite, finitely generated free or an orientable hyperbolic surface group. For a fixed element \(\gamma \in \Gamma\), a \(\gamma\)-random permutation in the symmetric group \(S_{N}\) is the image of \(\gamma\) through a uniformly random homomorphism \(\Gamma \rightarrow S_{N}\).\N\NIn the paper under review, the authors study local statistics of \(\gamma\)-random permutations and their asymptotics as \(N\) grows. They first consider \(\mathbb{E}[\mathrm{fix}_{\gamma}(N)]\), the expected number of fixed points in a \(\gamma\)-random permutation in \(S_{N}\). They show that, unless \(\gamma\) has finite order, the limit \(\mathbb{E}[\mathrm{fix}_{\gamma}(N)]\) when \(N \rightarrow \infty\) is an integer and is equal to the number of subgroups \(H \leq \Gamma\) such that \(\gamma \in H\) and \(H\simeq \mathbb{Z}\) or \(H= C_{2} \ast C_{2}\). Equivalently, this is the number of subgroups \(H \leq \Gamma\) containing \(\gamma\) and having (rational) Euler characteristic zero. Finally, the authors prove there is an asymptotic expansion for \(\mathbb{E}[\mathrm{fix}_{\gamma}(N)]\) and determine the limit distribution of the number of fixed points as \(N \rightarrow \infty\).