Orthospectra of Geodesic Laminations and Dilogarithm Identities on Moduli Space

Given a measured lamination on a finite area hyperbolic surface we consider a natural measure Mon the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure M gives summation identities for the Rogers dilogarithm function on the moduli space of a surface.

💡 Research Summary

The paper investigates a novel bridge between hyperbolic geometry, measured laminations, and special‑function identities on the moduli space of Riemann surfaces. Starting with a finite‑area hyperbolic surface Σ equipped with a measured lamination λ, the authors define an intersection function f_λ on the unit tangent bundle T¹Σ by assigning to each unit tangent vector the total transverse measure of λ intersected by the geodesic ray determined by that vector. Pushing the natural Liouville volume measure on T¹Σ forward through f_λ yields a measure M_λ on the real line. This construction captures the “orthospectrum” of λ – the collection of lengths of geodesic arcs that meet λ orthogonally – as the atomic part of M_λ, while the continuous part records more subtle intersection data.

The first major result establishes that M_λ is a finite measure whose total mass equals the hyperbolic area of Σ. By analyzing the moment generating function and Laplace transform of M_λ, the authors translate the discrete atomic contributions into a sum involving the Rogers dilogarithm function (\mathcal{R}(x)=\operatorname{Li}_2(x)+\frac12\log x\log(1-x)). Specifically, they prove the identity

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