Every word in a free group F induces a probability measure on every finite group in a natural manner. It is an open problem whether two words that induce the same measure on every finite group, necessarily belong to the same orbit of AutF. A special case of this problem, when one of the words is the primitive word x, was settled positively by the third author and Parzanchevski [PP15]. Here we extend this result to the case where one of the words is x d or [x, y] d for an arbitrary d ∈ Z.

Fix a word $ w $ in a free group $ \textbf {F}$ on $r$ generators. A $w$-random permutation in the symmetric group $S_{N}$ is obtained by sampling $r$ independent uniformly random permutations $ \sigma _{1},\ldots ,\sigma _{r}\in S_{N}$ and evaluating $w\left (\sigma _{1},\ldots ,\sigma _{r}\right )$. In [39, 40], it was shown that the average number of fixed points in a $w$-random permutation is $1+\theta \left (N^{1-\pi \left (w\right )}\right )$, where $ \pi \left (w\right )$ is the smallest rank of a subgroup $H\le \textbf {F}$ containing $w$ as a non-primitive element. We show that $ \pi \left (w\right )$ plays a role in estimates of all stable characters of symmetric groups. In particular, we show that for all $t\ge 2$, the average number of $t$-cycles is $ \frac {1}{t}+O\left (N^{-\pi \left (w\right )}\right )$. As an application, we prove that for every $s$, every $ \varepsilon>0$ and every large enough $r$, Schreier graphs with $r$ random generators depicting the action of $S_{N}$ on $s$-tuples, have 2nd eigenvalue at most $2\sqrt {2r-1}+\varepsilon $ asymptotically almost surely. An important ingredient in this work is a systematic study of not necessarily connected Stallings core graphs.