On the spectrum of twisted Laplacians and the Teichm\"uller representation

Abstract

We consider Laplacians with non unitary twists acting on sections of flat vector bundles over compact hyperbolic surfaces. These non self-adjoint Laplacians have discrete spectrum inside a parabola in the complex plane. For representations of the fundamental group of the base surface which are of Teichm\"uller type, we investigate the high energy limit and give a precise description of the bulk of the spectrum where Weyl's law is satisfied in terms of critical exponents of the representations which are completely determined by the Manhattan curve associated to the Teichm\"uller deformation. Our main result provides a counting estimate for the eigenvalues outside the bulk with a polynomial improvement over Weyl's law.