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

Let ₆ be a closed surface of genus g 2 and ₆ denote the fundamental group of ₆. We establish a generalization of Voiculescu's theorem on the asymptotic *-freeness of Haar unitary matrices from free groups to ₆. We prove that for a random representation of ₆ 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 ₆ is bounded as n. The proof involves an interplay between Dehn's work on the word problem in ₆ and classical invariant theory.