Divergence of Teichmueller Geodesics

TL;DR

This paper investigates the long-term behavior of Teichmueller geodesic rays, demonstrating divergence under specific conditions related to the transverse measures of their defining quadratic differentials.

Contribution

It establishes a new divergence criterion for Teichmueller geodesic rays based on the topological equivalence but non-absolute continuity of transverse measures.

Findings

Teichmueller geodesic rays diverge when transverse measures are topologically equivalent but not absolutely continuous.

The divergence depends on the measure-theoretic properties of the quadratic differentials.

Provides a criterion for divergence in the asymptotic geometry of Teichmueller space.

Abstract

We study the asymptotic geometry of Teichmueller geodesic rays. We show that when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, then the rays diverge in Teichmueller space.