This project is aimed at studying the geometry of subgroups of the mapping class group of a surface acting on the Teichmuller space and other related spaces. The guiding principal is an analogy between this action and the action of a Kleinian group on hyperbolic space. The primary topics studied by the PI are convex cocompactness (characterizations and examples), generalized combination theorems for Veech groups (geometric structure and ideal boundary behavior), and translation lengths for pseudo-Anosov mapping classes acting on Teichmuller space (especially its relation to minimal dilatation questions). Various parts of this work involve ongoing joint projects with R. Kent, B. Farb, and D. Margalit.
Homeomorphisms of a surface (which can be thought of as self-symmetries of the surface in a very weak sense) and the mapping class group (which is the collection of all such self-symmetries) are widely studied in mathematics. While these objects are primarily of interest in low dimensional topology and geometric group theory--two fields which have seen significant growth over the last 20 to 30 years--they also arise in a variety of other areas including complex analysis, algebraic geometry, and dynamics. For example, one of the most intriguing spaces in mathematics is the space which classifies surfaces equipped with a variety of geometric or algebraic structures (the so-called ``moduli space of Riemann surfaces''). This space is encoded in the Teichmuller space and the action of the mapping class group on it. The goal of this project is to develop a better understanding of certain classes of subgroups of the mapping class group through the geometry of their action on Teichmuller space.
