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

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.