Geodesic Envelopes in the Thurston Metric on Teichmuller space

Abstract

The Thurston metric on Teichmuller space, first introduced by W. P. Thurston is an asymmetric metric on Teichmuller space defined by $d_{Th}(X,Y) = \frac12 log\sup_{\alpha} \frac{l_{\alpha}(Y)}{l_{\alpha}(X)}$. This metric is geodesic, but geodesics are far from unique. In this thesis, we show that in the once-punctured torus, and in the four-times punctured sphere, geodesics stay a uniformly-bounded distance from each other. In other words, we show that the \textbf{width} of the geodesic envelope, E(X,Y) between any pair of points $X,Y \in T(S)$ (where $S = S_{1,1}$ or $S = S_{0,4}$) is bounded uniformly. To do this, we first identify extremal geodesics in Env(X,Y), and show that these correspond to stretch vectors, proving a conjecture of Huang, Ohshika and Papadopoulos. We then compute Fenchel-Nielsen twisting along these paths, and use these computations, along with estimates on earthquake path lengths, to prove the main theorem.