Random Unitary Representations of Surface Groups II: The large $n$ limit

TL;DR

This paper extends Voiculescu's theorem on asymptotic freeness of Haar unitary matrices from free groups to surface groups, showing bounded expected traces for large matrix sizes using geometric and algebraic techniques.

Contribution

It generalizes asymptotic freeness results to surface groups and connects geometric structures with random matrix theory.

Findings

Expected trace of fixed non-identity elements remains bounded as n→∞.

Generalization of Voiculescu's theorem to surface groups.

Uses geometric and algebraic methods involving moduli space and word problem.

Abstract

Let $\Sigma_{g}$ be a closed surface of genus $g\geq 2$ and $\Gamma_{g}$ denote the fundamental group of $\Sigma_{g}$. We establish a generalization of Voiculescu's theorem on the asymptotic $*$-freeness of Haar unitary matrices from free groups to $\Gamma_{g}$. We prove that for a random representation of $\Gamma_{g}$ into $\mathsf{SU}(n)$, with law given by the volume form arising from the Atiyah-Bott-Goldman symplectic form on moduli space, the expected value of the trace of a fixed non-identity element of $\Gamma_{g}$ is bounded as $n\to\infty$. The proof involves an interplay between Dehn's work on the word problem in $\Gamma_{g}$ and classical invariant theory.