On multiplicities in length spectra of semi-arithmetic hyperbolic surfaces
Mikhail Belolipetsky, Gregory Cosac, Cayo Dória, Gisele Teixeira Paula
TL;DR
This work proves that every cocompact and every non-cocompact semi-arithmetic Fuchsian group of arithmetic dimension $2$ that admits a modular embedding exhibits exponential growth of mean multiplicities (EGMM) in its length spectrum. The proofs split by case: the cocompact case leverages Schwarz-Pick bounds and arithmetic structure to obtain trace growth constraints, while the non-cocompact case reduces to a compact core via the generalized Kirszbraun extension and mirrors the same argument, aided by a weak bounded-clustering lemma. The paper also compiles concrete examples, notably triangle groups and Veech groups, with explicit lists and modular embeddings, illustrating the prevalence of EGMM beyond arithmetic groups. Finally, it discusses implications for quantum chaos and spectral statistics, linking EGMM to trace formulae and the expected behaviors of energy-level distributions, and notes open questions for semi-arithmetic groups with arithmetic dimension greater than two.
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.