Asymptotic properties of the quantum representations of the mapping class group

Abstract

For any surface with genus $\geqslant 2$, the monodromy of Hitchin’s connection is a projective representation of the mapping class group of the surface. We establish two results on the large level limit of these representations. First we prove that these projective representations lift to asymptotic representations. Second we show that under an infinitesimal rigidity assumption the characters of these representations have an asymptotic expansion. This proves the Witten’s asymptotic conjecture for mapping tori of surface diffeomorphisms. Our result is not limited to Seifert manifolds and applies to hyperbolic manifolds.