Geometric Group Theory and Surface Dynamics

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The proposal divides into three projects in the related fields of two dimensional dynamical systems and geometric group theory. The first is joint work with John Franks. One question that we address is : for which elements of the diffeomorphism group of a surface does the centralizer have a finite index subgroup with a global fixed point. This relates directly to understanding which subgroups of the mapping class group can act faithfully on a surface by diffeomorphisms. We are also looking for an analog in the group of area preserving homeomorphisms of the disk, of the Calabi invariant, which is defined for area preserving diffeomorphisms of the disk. The goal of the second project, which is a collaboration with Mark Feighn, is to solve the conjugacy problem for the group of outer automorphisms of a free group. The third project continues joint work with Lee Mosher. One of our goals is to show that an infinite subgroup of the outer automorphism group of a free group is either reducible or contains a fully irreducible element. Another is to show that the complex of free splittings of the free group is hyperbolic. The mapping class group of a surface, the diffeomorphism group of a surface and the outer automorphism group of the free group are related in fundamental ways. Classification theorems for elements of the mapping class group yield information about the algebraic properties of subgroups of the diffeomorphism group and so yield restrictions on the kinds of groups that can act on surfaces and on the ways in which groups can act. One of the main motivations for studying the outer automorphism group of the free group is its very close connection to mapping class groups. One part of the proposal focuses on finding global fixed points for subgroups of diffeomorphisms acting on a surface; i.e. points that are stationary for every element of the subgroup. Another is to understand actions of the mapping class group of a surface on that surface. Other parts of the proposal seek to generalize known important results about the mapping class group to the outer automorphism groupof the free group. Among these results are the conjugacy problem and a subgroup dichotomy. The former asks for an algorithm to decide if two elements differ only by a change of coordinates and the latter is the analogue for subgroups of a basic classification theorem for individual elements.