Weil--Petersson geodesics on the modular surface

TL;DR

This paper studies Weil--Petersson geodesics on the modular surface, revealing their geometric relation to hyperbolic geodesics, and shows how these relations lead to a homeomorphism with a singular measure on the circle of directions.

Contribution

It establishes a detailed geometric and measure-theoretic correspondence between Weil--Petersson and hyperbolic geodesics on the modular surface, including a homeomorphism with a singular measure.

Findings

WP and hyperbolic geodesics fellow-travel in the thick part.

Deviations occur during cusp excursions and are quantitatively analyzed.

The circle of directions admits a homeomorphism pushing forward Lebesgue measure to a singular measure.

Abstract

We consider the Weil--Petersson (WP) metric on the modular surface. We lift WP geodesics to the universal cover of the modular surface and analyse geometric properties of the lifts as paths in the hyperbolic metric on the universal cover. For any pair of distinct points in the thick part of the universal cover, we prove that the WP and hyperbolic geodesic segments that connect the pair, fellow-travel in the thick part and all deviations between these segments arise during cusp excursions. Furthermore, we give a quantitative analysis of the deviation during an excursion. We leverage the fellow traveling to derive a correspondence between recurrent WP and hyperbolic geodesic rays from a base-point. We show that the correspondence can be promoted to a homeomorphism on the circle of directions. By analysing cuspidal winding of a typical WP geodesic ray, we show that the homeomorphism pushes forward a Lebesgue measure on the circle to a singular measure. In terms of continued fraction coefficients, the singularity boils down to a comparison that we prove, namely, the average coefficient is bounded along a typical WP ray but unbounded along a typical hyperbolic ray.