2023/11/07 by Assaf Bar-Natan, Bar-Natan, Assaf
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2311.03709
openalex publication_date 2023/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Thurston metric on Teichmuller space, first introduced by W. P. Thurston is an asymmetric metric on Teichmuller space defined by dTh(X,Y) = \frac12 logsupα (lα(Y))/(lα(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 width of the geodesic envelope, E(X,Y) between any pair of points X,Y ∈ T(S) (where S = S1,1 or S = S0,4) 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.