Strong convergence of random representations of free products of finite groups

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 $C_2*C_3 \cong {\rm PSL}_2({\bf Z})$.