Asymptotic independence for random permutations from surface groups

Let $X$ be an orientable hyperbolic surface of genus $g\ge 2$ with a marked point $o$, and let $\mathrm{\Gamma }$ be an orientable hyperbolic surface group isomorphic to ${\pi }_1(X,o)$. Consider the space $\text{Hom}(\mathrm{\Gamma },S_n)$ which corresponds to $n$-sheeted covers of $X$ with labeled fiber. Given $\gamma \in \mathrm{\Gamma }$ and a uniformly random $\phi \in \text{Hom}(\mathrm{\Gamma },S_n)$, what is the expected number of fixed points of $\phi (\gamma )$? Formally, let $F_n(\gamma )$ denote the number of fixed points of $\phi (\gamma )$ for a uniformly random $\phi \in \text{Hom}(\mathrm{\Gamma },S_n)$. We think of $F_n(\gamma )$ as a random variable on the space $\text{Hom}(\mathrm{\Gamma },S_n)$. We show that an arbitrary fixed number of products of the variables $F_n(\gamma )$ are asymptotically independent as $n\to \mathrm{\infty }$ when there are no obvious obstructions. We also determine the limiting distribution of such products. Additionally, we examine short cycle statistics in random permutations of the form $\phi (\gamma )$ for a uniformly random $\phi \in \text{Hom}(\mathrm{\Gamma },S_n)$. We show a similar asymptotic independence result and determine the limiting distribution.