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 $\Gamma =G_1\ast \cdots \ast G_m$. Concretely, for a uniformly random ${\varphi }_N\in \mathrm{hom}(\Gamma ,\mathrm{Sym}(N))$, set ${\pi }_N=\mathrm{std}\circ {\varphi }_N$, where $\mathrm{std}$ denotes the standard $(N-1)$-dimensional representation of $\mathrm{Sym}(N)$ (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 $C_2\ast C_2\ast C_2$ and an explicit non-Ramanujan limiting spectral radius for $C_2\ast C_3\cong {\mathrm{PSL}}_2(\mathbf{Z})$.

38 pages, 1 figure