## Departmental PhD Thesis Exam – Assaf Bar-Natan

Tuesday, August 16, 2022 at 12:00 p.m. (sharp)

PhD Candidate: Assaf Bar-Natan
Supervisor: Kasra Rafi
Thesis title: Geodesic Envelopes in Teichmuller Space Equipped with the Thurston Metric

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The Thurston metric on Teichm\”{u}ller space, first introduced by W. P. Thurston is an asymmetric metric on Teichm\”{u}ller 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 \textit{width} of the \textit{geodesic envelope}, $E(X,Y)$ between any pair of points $X,Y \in \mc{T}(S)$ (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 \textit{stretch vectors}. 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.

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A draft of the thesis can be found here: thesis