The asymptotic cone of Teichmüller space and thickness

Sultan, Harold

doi:10.2140/agt.2015.15.3071

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The asymptotic cone of Teichmüller space and thickness

Harold Sultan

Algebraic & Geometric Topology 15 (2015) 3069–3104

DOI: 10.2140/agt.2015.15.3071

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.

Keywords

Teichmüller space, asymptotic cone, thickness

Mathematical Subject Classification 2010

Primary: 30F60, 20F69

Secondary: 20F65, 20F67

References

Publication

Received: 24 November 2014

Revised: 22 January 2015

Accepted: 2 February 2015

Published: 10 December 2015

Authors

Harold Sultan
Department of Mathematics Brandeis University Waltham, MA 02453 USA

Volume 15, issue 5 (2015)