Geodesic Envelopes in the Thurston Metric on Teichmuller space

The Thurston metric on Teichmuller space, first introduced by W. P. Thurston is an asymmetric metric on Teichmuller space defined by $d_{Th}(X,Y)=\frac{1}{2}log{\sup }_{\alpha }\frac{l_{\alpha }(Y)}{l_{\alpha }(X)}$. 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 $\mathbf{\text{width}}$ 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.