Notes on Stable Teichmuller quasigeodesics

TL;DR

This paper proves that cobounded Lipschitz paths in Teichmüller space with strongly relatively hyperbolic pullback bundles are close to geodesics, linking hyperbolic geometry properties with Teichmüller geodesics.

Contribution

It establishes a connection between the hyperbolic structure of pullback bundles and the approximation of paths by geodesics in Teichmüller space.

Findings

Cobounded Lipschitz paths with hyperbolic pullback bundles are close to geodesics.

Strong relative hyperbolicity of the pullback bundle implies geodesic approximation.

Provides new insights into the geometric structure of Teichmüller space.

Abstract

In this note, we prove that for a cobounded,Lipschitz path $\gamma:I\to\TT$, if the pull back bundle $\mathcal H_{\gamma}$ over $I$ is a strongly relatively hyperbolic metric space then there exists a geodesic $\xi$ in $\TT$ such that $\gamma(I)$ and $\xi$ are close to each other.