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

Abstract

Let ${\Sigma }_g$ be a closed surface of genus $g\ge 2$ and ${\Gamma }_g$ denote the fundamental group of ${\Sigma }_g$. We establish a generalization of Voiculescu's theorem on the asymptotic $\ast$-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.

version to appear in Geometry and Topology, 55 pages

References in corpus (3)

• Random Unitary Representations of Surface Groups I: Asymptotic expansions
• The Asymptotic Statistics of Random Covering Surfaces
• Large $N$ behaviour of the two-dimensional Yang-Mills partition function

Cited by in corpus (2)

• Random Unitary Representations of Surface Groups I: Asymptotic expansions
• The Asymptotic Statistics of Random Covering Surfaces