Convex structures of the unit tangent spheres in Teichm{ü}ller space

TL;DR

The paper provides a complete description of the convex geometry of the Thurston unit sphere in Teichmüller space by linking faces, exposed faces, and extreme points to chain-recurrent geodesic laminations and stretch vectors. It shows that every unit tangent can be written as a positive linear combination of stretch vectors along maximal laminations containing a core lamination, and that faces correspond to these cores via $F_\lambda$, with exposure tied to unmeasured laminations and extremality to maximal chain-recurrent laminations. This yields an affirmative characterization of extreme points and offers a new, cotangent-free route to infinitesimal rigidity and equilibration properties of stretch vectors under isometries. The results provide an effective framework for understanding the infinitesimal geometry of Thurston’s metric and its topological rigidity aspects in Teichmüller space.

Abstract

We analyse the convex structure of the Finsler infinitesimal balls of the Thurston metric on Teichm{ü}ller space. We obtain a characterisation of faces, exposed faces and extreme points of the unit spheres. In particular, we prove that with every face a unique chain-recurrent geodesic lamination is naturally associated in such a way that this face consists of unit tangent vectors expressed as linear combinations of stretch vectors along maximal geodesic laminations containing the given one. We show that this face is exposed if and only if its associated chain-recurrent geodesic lamination is an unmeasured lamination, that is, the support of a measured lamination. Furthermore, we show that a point on a unit tangent sphere is an extreme point if and only if it is a unit stretch vector along some maximal chain-recurrent geodesic lamination. The last gives an affirmative answer to a conjecture whose answer was known positively in the case where the surface is either the once-punctured torus or the 4-punctured sphere. Our main results give an alternative approach to the topological part of the infinitesimal rigidity of Thurston's metric and the equivariance property of stretch vectors.

Figures (1)

• Figure 1: The picture of the set-up in Lemma \ref{['flows']}

Theorems & Definitions (34)

• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• Theorem 2.5: Theorem 8.4 in ThM
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• proof