Word Measures on Wreath Products II

Understand

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 $FI_G$-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.