  Let $\alpha$ and $\beta$ be uniformly random permutations of orders $2$ and
$3$, respectively, in $S_{N}$, and consider, say, the permutation
$\alpha\beta\alpha\beta^{-1}$. How many fixed points does this random
permutation have on average? The current paper studies questions of this kind
and relates them to surprising topological and algebraic invariants of elements
in free products of groups.
  Formally, let $\Gamma=G_{1}*\ldots*G_{k}$ be a free product of groups where
each of $G_1,\ldots,G_k$ 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 $\Gamma\to S_{N}$. In this
paper we study local statistics of $\gamma$-random permutations and their
asymptotics as $N$ grows. We first consider
$\mathbb{E}\left[\mathrm{fix}_{\gamma}\left(N\right)\right]$, the expected
number of fixed points in a $\gamma$-random permutation in $S_{N}$. We show
that unless $\gamma$ has finite order, the limit of
$\mathbb{E}\left[\mathrm{fix}_{\gamma}\left(N\right)\right]$ as $N\to\infty$ is
an integer, and is equal to the number of subgroups $H\le\Gamma$ containing
$\gamma$ such that $H\cong\mathbb{Z}$ or $H\cong C_{2}*C_{2}$. Equivalently,
this is the number of subgroups $H\le\Gamma$ containing $\gamma$ and having
(rational) Euler characteristic zero. We also prove there is an asymptotic
expansion for $\mathbb{E}\left[\mathrm{fix}_{\gamma}\left(N\right)\right]$ and
determine the limit distribution of the number of fixed points as $N\to\infty$.
These results are then generalized to all statistics of cycles of fixed
lengths.
