Some Lipschitz maps between hyperbolic surfaces with applications to Teichmüller theory

Abstract

In the Teichmüller 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 cases of 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 particular hyperbolic pairs of pants and, more generally, between some hyperbolic surfaces of finite type with arbitrary genus and arbitrary number of boundary components. The Lipschitz-minimizing maps that we construct are distinct from Thurston’s stretch maps.