Strong convergence of random representations of free products of finite groups

TL;DR

This paper extends polynomial methods to analyze the strong convergence of random representations of free products of finite groups, revealing spectral properties and asymptotic behaviors of associated random graphs.

Contribution

It introduces fractional-power asymptotic expansions and verifies their applicability to free products of finite groups, establishing strong convergence results.

Findings

Proves strong convergence of random representations to the regular representation.

Derives asymptotically sharp spectral gaps for random Schreier graphs.

Identifies almost Ramanujan behavior and explicit spectral radii for specific free products.

Abstract

We extend the polynomial method of Chen--Garza-Vargas--Tropp--van Handel and Magee--Puder--van Handel for operator-norm bounds in random permutation models to the setting where torsion is present. The main new feature is that asymptotic expansion of traces naturally involves fractional powers of $N$ rather than an ordinary Laurent series. We formulate fractional-power analogues of the method's key hypotheses and prove they lead to strong convergence. We verify these analogues for free products of finite groups $\Gamma=G_1*\cdots*G_m$. Concretely, for a uniformly random $\phi_N\in{\rm hom}(\Gamma,{\rm Sym}(N))$, set $\pi_N = {\rm std} \circ \phi_N$, where ${\rm std}$ denotes the standard $(N-1)$-dimensional representation of ${\rm Sym}(N)$ (the permutation representation with the trivial subrepresentation removed). We deduce strong convergence of $\pi_N$ to the left regular representation of $\Gamma$. As applications, we obtain asymptotically sharp spectral gaps for the associated random Schreier graphs, including almost Ramanujan behavior for $C_2*C_2*C_2$ and an explicit non-Ramanujan limiting spectral radius for $C_2*C_3 \cong {\rm PSL}_2({\bf Z})$.