The asymptotic cone of Teichmüller space and thickness

Abstract

We study the asymptotic geometry of Teichmüller space equipped with the Weil–Petersson metric. In particular, we provide a characterization of the canonical finest pieces in the tree-graded structure of the asymptotic cone of Teichmüller space along the same lines as a similar characterization for right angled Artin groups and for mapping class groups. As a corollary of the characterization, we complete the thickness classification of Teichmüller spaces for all surfaces of finite type, thereby answering questions of Behrstock, Druţu and Mosher, and Brock and Masur. In particular, we prove that Teichmüller space of the genus-two surface with one boundary component (or puncture) is the only Teichmüller space which is thick of order two.