On multiplicities in length spectra of semi-arithmetic hyperbolic surfaces

TL;DR

This paper demonstrates that semi-arithmetic hyperbolic surfaces with modular embeddings exhibit exponential growth in average length spectrum multiplicities, extending known results from arithmetic surfaces and exploring their relation to quantum degeneracies.

Contribution

It establishes exponential growth of mean multiplicities in length spectra for semi-arithmetic surfaces with modular embeddings, a significant extension of prior arithmetic surface results.

Findings

Exponential growth of mean multiplicities in length spectra

Connection between length spectrum degeneracies and quantum system quantization

Extension of known results from arithmetic to semi-arithmetic 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.