Random Unitary Representations of Surface Groups I: Asymptotic Expansions

Abstract

AbstractIn this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let $$\Sigma _{g}$$ Σ g denote a topological surface of genus $$g\ge 2$$ g ≥ 2 . We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) under a random representation of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) into $$\mathsf {SU}(n)$$ SU ( n ) . Each such expected value involves a contribution from all irreducible representations of $$\mathsf {SU}(n)$$ SU ( n ) . The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.