Word maps and surface relations in symmetric groups

We study the expected number of fixed points of a random permutation obtained via a word map, with surface group constraints imposed. For stable irreducible characters $\chi$ of the symmetric group $S_{n}$ and with $R_{g}=[a_{1},b_{1}]\dots[a_{g},b_{g}]$ and $w\in F_{2g}$, we compute $\mathbb{E}_{S_{n}^{2g}}\left[\chi\left(R_{g}(h)\right)\#\mathrm{fix}\left(w(h)\right)\right]$. We show that, if $w$ is a shortest representative for the conjugacy class of $\gamma\in\Gamma_{g}=\left\langle a_{1},b_{1},\dots,a_{g},b_{g}:R_{g}\right\rangle$, then this expectation is $O\left(1/\dim\chi\right)$. As an application, we recover a boundedness statement of Magee--Puder on the large $n$ limit of the expected number of fixed points of $\phi_{n}(\gamma)$, where $\gamma\in\Gamma_{g}$ is fixed and $\phi_{n}\in\hom\left(\Gamma_{g},S_{n}\right)$ is chosen uniformly at random.