The Asymptotic Cone of Teichm\"uller Space: Thickness and Divergence

TL;DR

This paper characterizes the asymptotic cone of Teichmüller space with the Weil-Petersson metric, classifies its thickness, and analyzes divergence and contracting geodesics, advancing understanding of its large-scale geometry.

Contribution

It provides a canonical characterization of the asymptotic cone's structure, completes the thickness classification for all finite-type surfaces, and introduces a new complex of separating multicurves.

Findings

Teichmüller space of genus two with boundary is thick of order two.

It exhibits superquadratic but at most cubic divergence.

Strongly contracting quasi-geodesics are characterized in this setting.

Abstract

We study the Asymptotic Cone of Teichm\"uller 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 Teichm\"uller space along the same lines as a similar characterization for right angled Artin groups by Behrstock-Charney and for mapping class groups by Behrstock-Kleiner-Minksy-Mosher. As a corollary of the characterization, we complete the thickness classification of Teichm\"uller spaces for all surfaces of finite type, thereby answering questions of Behrstock-Drutu, Behrstock-Drutu-Mosher, and Brock-Masur. In particular, we prove that Teichm\"uller space of the genus two surface with one boundary component (or puncture) can be uniquely characterized in the following two senses: it is thick of order two, and it has superquadratic yet at most cubic divergence. In addition, we characterize strongly contracting quasi-geodesics in Teichm\"uller space, generalizing results of Brock-Masur-Minsky. As a tool, we develop a complex of separating multicurves, which may be of independent interest.