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 conjugacy problem 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.
In dynamical systems, one studies the long term behavior of an evolving system, sometimes given by repeating a single transformation of a space. Entropy is a measure of the complexity, or lack of predictability, of the system. Handel and Franks work in the interface of dynamical systems and low dimensional topology with emphasis on transformations of surfaces, particularly the two dimensional sphere. In the positive entropy case, the underlying structure of the transformation is well understood; much less is known in the entropy zero case. Franks and Handel have been filling in this gap using relative mapping class group techniques which allow one to simplify the transformations without losing critical information. They have completed a paper on the sphere and are current working on generalizations to surfaces of higher genus. Groups are algebraic objects that occur throughout mathematics. There are three related families of groups that are of great interest in geometric group theory, the branch of mathematics in which one applies geometric and topological methods to study these algebraic objects: the group of invertible square matrices of a given dimension, mapping class groups of surfaces and outer automorphism groups of finite rank free groups. Matrix groups are classical and mapping class groups have been intensely studied over the last forty years. Less is known about outer automorphism groups but interest in this subject has grown over the last twenty-five years and accelerated in the last five. One important technique is to view elements of a given group as symmetries of a metric space. For example, the group of integers can be thought of as symmetries of a line: shifting left or right according to the given integer. This technique is especially useful when the space in question satisfies a property called hyperbolicity. There are several metric spaces associated with the outer automorphism group of a free group but until recently none were known to be hyperbolic. Handel and Mosher proved that one of these, the free splitting complex, is hyperbolic. Handel and Mosher also completed a series of four papers on subgroups of outer automorphism groups. These should be viewed as assembling a tool kit for future applications.
