Geometry, topology and group theory of surfaces

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AbstractAward: DMS-0905748Principal Investigator: Christopher J. LeiningerThis project aims to study the geometry of various structures onsurfaces, actions of the mapping class group on these spaces, andtopological/dynamical aspects of surface homeomorphisms. Thisincludes (1) an ongoing project with R.P. Kent on convexcocompactness in the mapping class group, with a focus on freegroups and surface groups; (2) a topological investigations ofalgebraic relations in the mapping class group and the closelyrelated braid groups with D. Margalit; (3) a study, via geodesiclength functions, of certain singular euclidean geometricstructures on surfaces and their degenerations in a joint projectwith M. Duchin and K. Rafi; (4) a continuing project withB. Farb and D. Margalit to provide a topological model for"minimal complexity" surface homeomorphisms.Surfaces---like the surface of a ball or a doughnut---have beenstudied for hundreds of years, and are fundamental and beautifulobjects in mathematics. The study of surfaces is intrinsicallyinteresting, but is also responsible for the creation of entirefields of mathematics, as well as the development of techniquesin many others. As such, the theory of surfaces and theirgeometries lies at the juncture of several fields of mathematicsincluding complex analysis, differential geometry,low-dimensional topology, geometric group theory and dynamics.Many geometric objects can be described using surfaces as thebasic building blocks. To study these objects one naturallyencounters the notion of a "homeomorphism" of a surface: this isa kind of "symmetry" which preserves only the most basicproperties of the surface. The set of all homeomorphisms issomewhat unwieldy, but the most important features can bedistilled into a more manageable structure called the "mappingclass group." This project proposes the study of several relatedproblems about surfaces, their homeomorphisms and mapping classgroups, and the implications of these studies to relatedgeometric objects.