Geometry and the mapping class group

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This project is aimed at studying the geometry of subgroups of the mappingclass group of a surface acting on the Teichmuller space and other relatedspaces. The guiding principal is an analogy between this action and theaction of a Kleinian group on hyperbolic space. The primary topicsstudied by the PI are convex cocompactness (characterizations andexamples), generalized combination theorems for Veech groups (geometricstructure and ideal boundary behavior), and translation lengths forpseudo-Anosov mapping classes acting on Teichmuller space (especially itsrelation to minimal dilatation questions). Various parts of this workinvolve ongoing joint projects with R. Kent, B. Farb, and D. Margalit.Homeomorphisms of a surface (which can be thought of as self-symmetries ofthe surface in a very weak sense) and the mapping class group (which isthe collection of all such self-symmetries) are widely studied inmathematics. While these objects are primarily of interest in lowdimensional topology and geometric group theory--two fields which haveseen significant growth over the last 20 to 30 years--they also arise in avariety of other areas including complex analysis, algebraic geometry, anddynamics. For example, one of the most intriguing spaces in mathematicsis the space which classifies surfaces equipped with a variety ofgeometric or algebraic structures (the so-called ``moduli space of Riemannsurfaces''). This space is encoded in the Teichmuller space and theaction of the mapping class group on it. The goal of this project is todevelop a better understanding of certain classes of subgroups of themapping class group through the geometry of their action on Teichmullerspace.