On multiplicities in length spectra of semi-arithmetic hyperbolic surfaces

Abstract

We show that semi-arithmetic surfaces of arithmetic dimension two which admit a modular embedding have exponential growth of mean multiplicities in their length spectrum. Prior to this work large mean multiplicities were rigorously confirmed only for the length spectra of arithmetic surfaces. We also discuss the relation of the degeneracies in the length spectrum and quantization of the Hamiltonian mechanical system on the surface.