Strong convergence of uniformly random permutation representations of surface groups

TL;DR

The paper proves that uniformly random permutation representations of a surface group $\Gamma_g$ converge strongly in probability to the regular representation as the permutation degree $n$ grows, by developing a generalized polynomial method that handles non-rational spectral statistics. Central to the approach are (i) an effective $1/n$-expansion for expected traces via a Gevrey-type criterion, (ii) an effective polynomial/rational approximation for fixed-point statistics from random $S_n$-embeddings, and (iii) a hyperbolic-geometry analysis of proper powers to control random-walk growth. These tools yield a polynomial-rate strong convergence, and their consequences imply that a random degree-$n$ cover of a closed hyperbolic surface typically has near-optimal spectral gap, ignoring base eigenvalues. The results extend the polynomial method beyond rational settings and provide a robust framework for strong convergence in geometric group contexts, with implications for spectral theory on random covers and variable negative curvature surfaces.

Abstract

Let $Γ$ be the fundamental group of a closed orientable surface of genus at least two. Consider the composition of a uniformly random element of $\mathrm{Hom}(Γ,S_n)$ with the $(n-1)$-dimensional irreducible representation of $S_n$. We prove the strong convergence in probability as $n\to\infty$ of this sequence of random representations to the regular representation of $Γ$. As a consequence, for any closed hyperbolic surface $X$, with probability tending to one as $n\to\infty$, a uniformly random degree-$n$ covering space of $X$ has near optimal relative spectral gap -- ignoring the eigenvalues that arise from the base surface $X$. To do so, we show that the polynomial method of proving strong convergence can be extended beyond rational settings. To meet the requirements of this extension we prove two new kinds of results. First, we show there are effective polynomial approximations of expected values of traces of elements of $Γ$ under random homomorphisms to $S_n$. Secondly, we estimate the growth rates of probabilities that a finitely supported random walk on $Γ$ is a proper power after a given number of steps.

Figures (3)

• Figure 5.1: Illustration of the tiling $\mathcal{T}$ and embedded Cayley graph of $\Gamma$ for $g=2$.
• Figure 5.2: Illustration of Fact \ref{['fact:about Chet-shape']} (half-plane model of $\mathbb{H}^{2}$)
• Figure 5.3: Illustration of Definition \ref{['def:prod-path']} and Lemma \ref{['lem:in H^2 prod and arc are close']}

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.5
• Theorem 1.6: MPasympcover
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3