[Paper Review] The Asymptotic Cone of Teichm\"uller Space: Thickness and Divergence

Harold Mark Sultan|arXiv (Cornell University)|Nov 28, 2012

Geometric and Algebraic Topology38 references4 citations

TL;DR

This paper characterizes the canonical finest pieces in the tree-graded structure of the asymptotic cone of Teichm"uller space under the Weil-Petersson metric, using a newly defined complex of separating multicurves. It proves that Teichm"uller space for the genus two surface with one boundary component (S2,1) is thick of order two and has superquadratic yet at most cubic divergence, uniquely distinguishing it among all finite-type surfaces.

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.

Motivation & Objective

• To characterize the canonical finest pieces in the tree-graded structure of the asymptotic cone of Teichm"uller space with the Weil-Petersson metric.
• To complete the thickness classification of Teichm"uller spaces for all finite-type surfaces, resolving open questions by Behrstock-Drutću, Behrstock-Drutću-Mosher, and Brock-Masur.
• To provide a complete divergence classification of Teichm"uller spaces, identifying the unique case of superquadratic yet at most cubic divergence.
• To generalize results on strongly contracting quasi-geodesics in Teichm"uller space, extending prior work by Brock-Masur-Minsky.
• To introduce and study a new complex of separating multicurves as a key technical tool for analyzing the asymptotic cone structure.

Proposed method

• Define a new complex of separating multicurves that encodes product structures in the pants complex and relates to the asymptotic cone's tree-graded decomposition.
• Use ultralimits and asymptotic cones to analyze the large-scale geometry of Teichm"uller space, focusing on cut-points and separation properties.
• Establish equivalence between two points lying in the same finest piece of the asymptotic cone and the existence of sequences with uniformly bounded distance in the surface's marking complex.
• Apply geometric and combinatorial arguments involving quasi-geodesics, projections, and path constructions in the asymptotic cone to derive contradiction-based proofs.
• Leverage the relationship between thickness and divergence, using Theorem 5.21 to bound divergence polynomially by degree of thickness plus one.
• Use results from the pants complex and its quasi-isometry to Teichm"uller space to transfer coarse geometric properties from the combinatorial complex to the metric space.

Experimental results

Research questions

• RQ1 What characterizes the canonical finest pieces in the tree-graded structure of the asymptotic cone of Teichm"uller space under the Weil-Petersson metric?
• RQ2 For which surfaces is Teichm"uller space thick of order two, and what geometric properties distinguish such spaces?
• RQ3 What is the precise divergence function of Teichm"uller space for the genus two surface with one boundary component?
• RQ4 How do the asymptotic cone's cut-point structure and tree-graded decomposition relate to the geometry of separating multicurves?
• RQ5 Can the characterization of strongly contracting quasi-geodesics in Teichm"uller space be generalized beyond previous results?

Key findings

• The canonical finest pieces in the asymptotic cone of Teichm"uller space are characterized by the existence of representative sequences with uniformly bounded distance in the marking complex.
• Teichm"uller space of the genus two surface with one boundary component (S2,1) is thick of order two, uniquely distinguishing it among all finite-type surfaces.
• The divergence of Teichm"uller space for S2,1 is superquadratic yet at most cubic, providing the first known example of such divergence in this context.
• The asymptotic cone of Teichm"uller space for S2,1 contains nontrivial flats and is tree-graded, with cut-points present in the cone, but not all points are global cut-points.
• The complex of separating multicurves is introduced and shown to be instrumental in analyzing the asymptotic cone's structure and the thickness classification.
• The paper completes the thickness and divergence classification of Teichm"uller spaces for all finite-type surfaces, with S2,1 being the only surface with superquadratic yet at most cubic divergence.