The combinatorial structure of the unit tangent spheres and cotangent spheres of Teichm{\"u}ller space with Thurston's Finsler metric

Abstract

We prove several new results on the combinatorial structures of the unit spheres of the norms induced by Thurston's metric on the tangent and cotangent spaces of the Teichm{\"u}ller space of a closed surface of negative Euler characteristic. These results include a formula for the dimension of every face of a unit sphere in the tangent space in terms of an invariant of the chain-recurrent lamination representing the face. We then prove that the combinatorial structure of such a unit sphere is independent of the underlying point in Teichm{\"u}ller space. Provided the genus of the surface is $\ge$ 2, we show that there is a natural isomorphism between the extended mapping class group of the surface and the group of combinatorial automorphisms of such a unit sphere. In the case of genus 2, we obtain a natural epimorphism between the two groups whose kernel is the class of the hyperelliptic involution. Regarding the unit spheres of Thurston's metric in the cotangent spaces, we obtain a formula describing the codimensions of faces of such a sphere in terms of corresponding projective measured laminations. We then give a necessary and sufficient condition for a face to be exposed, and of a face to correspond to a projectively weighted multi-curve. Some of the results obtained answer open questions.