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.

arXiv commentversion to appear in Geometry and Topology, 55 pages

References in corpus (6)

• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Matrix Group Integrals, Surfaces, and Mapping Class Groups I: $U(n)$
• Random Unitary Representations of Surface Groups I: Asymptotic expansions
• The Asymptotic Statistics of Random Covering Surfaces
• Word Measures on Unitary Groups
• 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