Geometric Group Theory and Surface Dynamics

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There are three sections to the proposal, each the continuation of a long standing collaboration. Lee Mosher and the PI have short, medium and long term goals regarding the outer automorphism group Out(F) of the finite rank free group F. These include proving that subgroups of Out(F) are either virtually abelian or have infinite dimensional second bounded cohomology and developing an analog for Out(F) of the Masur-Minsky summation formula that applies to mapping class groups. Mark Feighn and the PI have been focusing on the conjugacy problem for Out(F) in the case that the elements in question have polynomial growth. The intent is to complete this important special case and then to the solve the full conjugacy problem by merging the techniques from the polynomially growing case with those already developed to solve the exponentially growing case. John Franks and the PI use relative mapping class group methods to study the dynamics of diffeomorphisms of surfaces. One goal is to to extend their work on zero entropy area preserving diffeomorphisms of the sphere to surfaces of higher genus. This project concerns the properties of, and the relationships between, three important groups: the outer automorphism group of a free group, the mapping class group of a surface, and the diffeomorphism group of a surface. Topological and geometric methods are used to study dynamical systems and in return, ideas from dynamics are used to formulate and prove fundamental topological and geometric properties of outer automorphisms and mapping classes. There is currently a great deal of interest in the large scale geometry of the group of outer automorphisms of the free group and this proposal is well situated to make substantial progress on this front.