Coarse and Fine Geometry of the Thurston Metric

Abstract

We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface $S$ . Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem.

Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space

“…This tells us that the stretch map id S : (S, h 0 ) → (S, h s ) composed with the stretch map id S : (S, h s ) → (S, h s+t ) is precisely equal to the stretch map id S : (S, h 0 ) → (S, h s+t ). Therefore, we see that the equivalence class of K(h s , h s+t ) = t for all s, t ≥ 0, (6) and hence the equivalence classes of (S, h t ), for t ≥ 0, form a geodesic ray for the curve metric K. Finally, the Lipschitz metric is at least the curve metric. Thus, we have…”

“…This (informal) non-hyperbolicity of the Thurston metric (Theorem 5.2) essentially comes from the fact that the envelope from X to Y in T(S 2 ), i.e. the union of all the geodesics from X to Y (a notion studied in [6]), becomes very "fat" as the distance between X and Y increases. Indeed, the geodesics G and G XY constructed in the proof of Theorem 5.2 fail to fellow-travel.…”

The infinitesimal and global Thurston geometry of Teichm{ü}ller space

“…Theorem 1.8 is key to our strategy for proving infinitesimal rigidity of the Thurston metric, and relies upon Ivanov's theorem [12,15,18]. This argument fails when (g, n) = (1, 1) or (0, 4), which are precisely the two cases are (effectively) covered by [6]. 1.7.…”

“…• the role of the finitely-constrained feasible region is served by the infinitely-constrained envelope Env(x, y) T(S) (see [6]) comprised of points on Thurston geodesics from x to y; • just as the feasible region is convex, one naturally expects Env(x, y) to be geodesically convex with respect to the Thurston metric; • the point x is the chosen initial basic feasible solution (i.e., extreme point), and the goal is to maximise the Thurston metric distance from x -the maximum distance is uniquely attained by y; • one expects the edges of Env(x, y) to be given by stretch paths with respect to maximal chain-recurrent laminations, and so we start at x and traverse along arbitrary stretch paths whilst increasing the distaince from x, and occasionally pivoting to new basic feasible solutions until we reach y.…”

“…The set of stretch vectors in S x of stretch paths with respect to maximal chain-recurrent laminations characterise the set of extreme points of S x . Conjecture 1.12 is true when S is either the once-punctured torus or the 4-punctured sphere [6], but is unknown for surfaces of general type. In fact, neither direction of the characterisation is known.…”