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. Each project is with a different coauthor and builds on previous joint work. The first is a collaboration with John Franks. One goal is to find a structure theorem for entropy zero, area preserving diffeomorphism of surfaces. Another is to decide if a finite index subgroup of the mapping class group of a surface of sufficiently high genus can act faithfully on a surface by area preserving diffeomorphisms. The goal of the second project, which is a collaboration with Mark Feighn, is to provide a complete solution to the conjugacyproblem for the outer automorphism group of the free group. The third project, which is a collaboration with Lee Mosher, also addresses fundamental properties of the outer automorphism group of the free group; the goals are to prove a relative version of the subgroup classification theorem and to develop a heirarchy theory.The proposal makes use of, and further develops, the deep connections between the mapping class group of a surface, the diffeomorphism group of a surface and the outer automorphism group of the free group. A great deal is known about positive entropy surface diffeomorphisms. One part of the proposal is to develop a structure theorem for zero entropy, area preserving surface diffeomorphisms. This work makes use of relative mapping class group techniques and has applications to the existence of actions of finite index subgroups of mapping class groups on surfaces. Other parts of the proposal seek to generalize known important results about the mapping class group to the outer automorphism group of the free group. Among these results are the conjugacy problem, a relative version of the subgroup classification theorem and a heirarchy theory: the first asks for an algorithm that decides if two elements differ only by a change of coordinates; the second is a complete analogue for subgroups of a basic classification theorem for individual elements; the third explores the geometry of various complexes and spaces associated to the outer automorphism group.