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RESEARCH STATEMENT MAPPING CLASS GROUPS IN THEIR NATURAL HABITAT

For an oriented topological surface S, a mapping class is the isotopy class of an orientationpreserving homeomorphism from S to itself. These constitute a discrete, infinite group known as Mod(S), the mapping class group of S. While inviting comparison to linear groups, Kleinian groups, Gromov-hyperbolic groups, and the group Out(Fn) consisting of outer automorphisms of the rank-n free group, mapping class groups have special relevance to geometric topology. In particular, mapping classes and their relations give direct information about the geometry of 3-manifolds [40] and the topology of 4-manifolds [3]. In the study of complex structures on surfaces, known as Teichmüller theory, mapping class groups appear as orbifold fundamental groups of moduli spaces. Investigation of mapping class groups draws from representation theory, dynamics, and geometric group theory tools such as coarse geometry and group actions on Gromov-hyperbolic or CAT(0) spaces. Of the myriad methods and motivations with which one may study mapping class groups, my research favors geometric insight and native settings. I am interested in what and how group-theoretic properties of mapping class groups stem from features of surfaces. These features include the geometry of Teichmüller space, and the combinatorics of simple closed loops on a surface, encoded in its curve complex. In what follows, I describe my previous results and outline a plan for ongoing research.