Mapping Class Groups, Teichmuller Spaces and Low Dimensional Topology

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AbstractProposal: DMS-9803497Principal Investigator: Howard MasurMasur will work on several topics involving the mapping class group andTeichmuller theory. He will finish his joint work with Yair Minsky onthe combinatorial group theory of the mapping class group. This workincludes solving the conjugacy problem and finding a biautomaticstructure. The method has been to use the hyperbolic structure of thecomplex of curves. Masur will begin a study of the large scale orasymptotic geometry of the mapping class group. This study is relatedto the study of the quasi-isometries of Teichmuller space. Masur willcontinue his work on the representation theory of the mapping classgroup. He will study certain functions called conditional negativekernels, with the goal of proving that the group does not have Kazhdanproperty T. He will also study whether surface groups admithomeomorphisms to the mapping class group, which is related to thestudy of surface bundles over surfaces, a problem in 4-manifoldtheory.The field of surface topology is the study of those properties ofsurfaces that do not change under deformations. The field interactsstrongly with other disciplines within mathematics, such as complexanalysis, and with outside disciplines such as physics. The mappingclass group, which is the group of self mappings of the surface, is afundamental object of study. Many mathematicians over the years havestudied the group from a variety of viewpoints. Recently, powerfulnew geometric tools have been introduced to study groups, which arethe objects of symmetry in mathematics. The principle investigator isworking to bring these new geometric ideas into the study of surfacesand the mapping class group.