Word Measures on Wreath Products II

Every word $w$ in $F_r$, the free group of rank $r$, induces a probability measure (the $w$-measure) on every finite group $G$, by substitution of random $G$-elements in the letters. This measure is determined by its Fourier coefficients: the $w$-expectations $E_w[\chi ]$ of the irreducible characters of $G$. For every finite group $G$, every stable character $\chi$ of $G\wr S_n$ (trace of a finitely generated $F{I}_G$-module), and every word $w\in F_r$, we approximate $E_w[\chi ]$ up to an error term of $O(n^{-\pi (w)})$, where $\pi (w)$ is the primitivity rank of $w$. This generalizes previous works by Puder, Hanany, Magee and the author. As an application we show that random Schreier graphs of representation-stable actions of $G\wr S_n$ are close-to-optimal expanders. The paper reveals a surprising relation between stable representation theory of wreath products and not-necessarily connected Stallings core graphs.