Some Lipschitz maps between hyperbolic surfaces with applications to Teichm\"uller theory

TL;DR

This paper constructs special geodesic lines in Teichmüller space using Lipschitz maps between hyperbolic surfaces, revealing new properties of stretch lines and their relation to Lipschitz constants.

Contribution

It introduces Lipschitz-minimizing maps between hyperbolic surfaces that differ from Thurston's stretch maps, with applications to Teichmüller theory.

Findings

Constructed geodesic lines in Teichmüller space that are symmetric under reversal.

Developed Lipschitz maps with controlled constants between hyperbolic hexagons and surfaces.

Provided explicit descriptions of these maps in Fenchel-Nielsen coordinates.

Abstract

In the Teichm\"uller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to reparametrization). The lines we construct are special stretch lines in the sense of Thurston. They are directed by complete geodesic laminations that are not chain-recurrent, and they have a nice description in terms of Fenchel-Nielsen coordinates. At the basis of the construction are certain maps with controlled Lipschitz constants between right-angled hyperbolic hexagons having three non-consecutive edges of the same size. Using these maps, we obtain Lipschitz-minimizing maps between hyperbolic particular pairs of pants and, more generally, between some hyperbolic sufaces of finite type with arbitrary genus and arbitrary number of boundary components. The Lipschitz-minimizing maps that we contruct are distinct from Thurston's stretch maps.