On multiplicities in length spectra of semi-arithmetic hyperbolic surfaces

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.

💡 Research Summary

The paper investigates the statistical behavior of closed geodesic lengths on hyperbolic surfaces arising from semi‑arithmetic Fuchsian groups. The authors focus on groups whose invariant trace field has arithmetic dimension two and which admit a modular embedding. Their main result is that for any such surface the mean multiplicity of lengths—i.e., the average number of distinct closed geodesics sharing the same length—grows exponentially with the length. Formally, there exists a constant (C>0) such that (\langle g(\ell)\rangle\ge e^{C\ell}) for all sufficiently large (\ell). This property, called EGMM (Exponential Growth of Mean Multiplicities), was previously proved only for arithmetic groups (arithmetic dimension one).

The authors begin by recalling the prime geodesic theorem and defining the multiplicity function (g(\ell_n)) for the ordered length set ({\ell_n}). They introduce the bounded‑clustering (B‑C) property, which controls the number of trace values in short intervals, and note that Sarnak conjectured B‑C characterizes arithmetic groups. While B‑C is known for arithmetic groups, the paper shows that a weaker version suffices for semi‑arithmetic groups of dimension two with a modular embedding.

A semi‑arithmetic group (\Gamma) is defined by a totally real trace field (K) and the integrality of traces of (\Gamma^{(2)}). The arithmetic dimension (r) counts the number of real embeddings of (K) for which the trace set is bounded; (r=1) corresponds to arithmetic groups. For (r=2), (\Gamma^{(2)}) can be embedded into (\mathrm{SL}(2,\mathbb R)^2) via two real embeddings (\rho_1,\rho_2). A modular embedding is a holomorphic (or anti‑holomorphic) map (\tilde F:\mathbb H\to\pm\mathbb H\times\pm\mathbb H) that intertwines the action of (\Gamma^{(2)}) with its image under the embedding. This yields a totally geodesic immersion of the surface (S=\Gamma^{(2)}\backslash\mathbb H) into the product space (\Delta\backslash(\pm\mathbb H\times\pm\mathbb H)).

The technical heart of the paper lies in Lemma 3.1, where the Schwarz–Picard lemma is applied to (\tilde F). Because (\tilde F) is not an automorphism, its derivative is uniformly bounded away from 1: (\sup_{z\in\mathbb H}|D\tilde F(z)|=1-\delta) for some (0<\delta<1). This bound translates into an inequality for the non‑trivial embedding (\sigma) of the trace field: \