Prescribing the behavior of Weil-Petersson geodesics in the moduli space of Riemann surfaces

TL;DR

This paper develops techniques to control and prescribe the behavior of Weil-Petersson geodesics in the moduli space of Riemann surfaces, including examples of closed and divergent geodesics with specific properties.

Contribution

It introduces new methods for controlling length-functions and twist parameters along WP geodesics, and constructs examples of geodesics with prescribed itineraries in the moduli space.

Findings

Existence of closed WP geodesics in the thin part of moduli space.

Construction of divergent WP geodesic rays with minimal filling ending lamination.

Development of a symbolic coding for laminations using subsurface coefficients.

Abstract

We study Weil-Petersson (WP) geodesics with narrow end invariant and develop techniques to control length-functions and twist parameters along them and prescribe their itinerary in the moduli space of Riemann surfaces. This class of geodesics is rich enough to provide for examples of closed WP geodesics in the thin part of the moduli space, as well as divergent WP geodesic rays with minimal filling ending lamination. Some ingredients of independent interest are the following: A strength version of Wolpert's Geodesic Limit Theorem proved in Sec.4. The stability of hierarchy resolution paths between narrow pairs of partial markings or laminations in the pants graph proved in Sec.5. A kind of symbolic coding for laminations in terms of subsurface coefficients presented in Sec.7.