Summary of Papers

The papers described here are grouped via topic and, within each group, listed in the order in which they were written. 1. The coarse geometry of Teichmüller space with the Teichmüller metric (1) The augmented marking complex of a surface. 2013; to appear in the Journal of the London Mathematical Society. I build an augmentation of the Masur-Minsky marking complex by Groves-Manning combinatorial horoballs to obtain a graph I call the augmented marking complex, AMpSq. Adapting work of Masur-Minsky, I show this augmented marking complex is quasiisometric to Teichmüller space T pSq with the Teichmüller metric, and also completely integrate the Masur-Minsky hierarchy machinery to AMpSq to build flexible families of uniform quasigeodesics in Teichmüller space, called hierarchy paths. As an application, I give a new proof of Rafi’s distance formula for T pSq with the Teichmüller metric. The Masur-Minsky marking complex was a central tool in the proofs of the geometric rank and quasiisometric rigidity theorems for the mapping class group. The augmented marking complex has played the same role, appearing in every proof of the corresponding theorems for the Teichmüller metric. While a similar construction was simultaneously and independently discovered by Eskin-Masur-Rafi, this is the only formal account of this machinery in the literature. (2) Elliptic actions on Teichmüller space. 2014; to appear in Groups, Geometry, and Dynamics. Let H ă MCGpSq be a finite subgroup of the mapping class group and consider the convex subset of T pSq fixed by H, FixpHq Ă T pSq, which Kerckhoff famously proved is always nonempty. In this paper, I study the geometry of the set of almost fixed points with respect to the Teichmüller metric. For any R ą 0, I prove that the set of points whose H-orbits have diameter bounded by R, FixRpHq, is contained in a bounded neighborhood of FixpHq. Roughly, this says that almost fixed points are close to fixed points. As an application, I show that the orbit of any point in T pSq has a fixed coarse barycenter, a property satisfied by any metric space of coarse nonpositive curvature, e.g. hyperbolic or CAT(0). By contrast, I build an explicit family of examples showing that FixRpHq need not be quasiconvex, despite the fact that FixpHq is convex. As an application of the barycenter theorem, I prove that there is an exponential-time algorithm to solve the conjugacy problem for finite order subgroups of MCGpSq, recovering a theorem of Tao. The main tools are the combinatorial machinery I described in the previous paper, and finer tools from Teichmüller and Weil-Petersson geometry. These results help with issues related to torsion in the semihyperbolicity project below.