Random unitary representations of surface groups, II : The large n limit

Let Σg be a closed surface of genus g≥ 2 and Γg denote the fundamental group of Σg. We establish a generalization of Voiculescu's theorem on the asymptotic $*$-freeness of Haar unitary matrices from free groups to Γg. We prove that for a random representation of Γg into 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 Γg is bounded as n→∞. The proof involves an interplay between Dehn's work on the word problem in Γg and classical invariant theory.