2012 Theses Doctoral

The Asymptotic Cone of Teichmuller Space: Thickness and Divergence

Sultan, Harold Mark

Using the geometric model of the pants complex, we study the Asymptotic Cone 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 similar characterizations for right angled Artin groups by Behrstock-Charney and for mapping class groups by Behrstock-Kleiner-Minsky-Mosher.

As a corollary of the characterization, we complete the thickness classification of Teichmüller spaces for all surfaces of finite type. In particular, we prove that Teichmüller 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 Teichmüller space, generalizing results of Brock-Masur-Minsky. As a tool in the thesis, we develop a natural relative of the curve complex called the complex of separating multicurves which may be of independent interest.

The final chapter includes various related and independent results including, under mild hypotheses, a proof of the equivalence of wideness and unconstrictedness in the CAT(0) setting, as well as adapted versions of three preprints. Specifically, in the three preprints we characterize hyperbolic type quasi-geodesics in CAT(0) spaces, we prove that the separating curve complex of the genus two surface satisfies a quasidistance formula and is Gromov-hyperbolic, and we study the net of separating pants decompositions in the pants complex.