Word Measures on Wreath Products II
Yotam Shomroni
TL;DR
This work develops a comprehensive framework linking word measures on wreath products $G\wr S_n$ to stable representation theory, via the algebra $\mathcal{A}(G)$ of stable class functions and a universal Induction-Convolution mechanism. It proves a unified asymptotic formula for $\mathbb{E}_w[f]$ for non-powers $w$, decomposing contributions from stable characters $\chi_\phi$ according to the primitivity rank $\pi(w)$, and extends to powers $w=u^k$ with refined behavior. The main technical tool, the Induction-Convolution Lemma, disentangles induction from convolution across $G\wr S_n$, enabling precise $O(n^{-\,\pi(w)})$-error estimates and enabling applications to random Schreier graphs. The results generalize prior works and establish near-optimal expansion bounds for Schreier graphs arising from rep-stable actions of wreath products, revealing deep connections between stable representation theory and Stallings core-graphs. This advances the understanding of word measures in finite groups and their spectral/expansion consequences in combinatorial and geometric group theory.
Abstract
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 $E_w[χ]$ of the irreducible characters of $G$. For every finite group $G$, every stable character $χ$ of $G\wr S_n$ (trace of a finitely generated $FI_G$-module), and every word $w\in F_r$, we approximate $E_w[χ]$ up to an error term of $O(n^{-π(w)})$, where $π(w)$ 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.