Geometric Group Theory and Surface Dynamics

Research Foundation Of The City University Of New York (Lehman), Bronx NY

Investigators

Abstract

Abstract Award: DMS-0103435 Principal Investigator: Michael Handel The proposal divides into four projects in the related fields of two dimensional dynamical systems and geometric group theory. The goal of the first, which is a collaboration with Mark Feighn, is to compute, in a transparent, geometric and algorithmic way, the centralizer of an element of the outer automorphism group. The second is to continue the principal investigator's study of the forcing order on braid types in the four times punctured disk. This project has both an experimental and theoretical part. Computer programs are used to generate examples and to aid in the formation of conjectures. Once conjectures are made that cannot be disproved by computer, rigorous proofs will be attempted. The third project is a collaboration with John Franks with the long term goal of proving that generic area preserving diffeomorphisms of the two dimensional sphere have dense periodic orbits. The more immediate goal is to find analogs for generic area preserving diffeomorphisms of the topological structure that is known to exist for twist maps. The fourth is a collaboration with Lee Mosher. The first steps in the project will be to identify and study quasi-lines that can play the role in Culler Vogtmann space that Teichmuller geodesics play in Teichmuller space. This proposal is concerned with the interface between two areas of mathematics: two dimensional dynamical systems and geometric group theory. The former studies the long term behavior of systems that evolve over time, while the latter treats algebraic objects by geometric means. The fields have been intertwined for more than fifty years and have been the focus of a great deal of research in the past twenty. Part of the proposal focuses on two long standing fundamental questions in two dimensional dynamical systems. The first examines how simple systems change into chaotic ones. The second concerns transformations of the sphere that preserve area, and asks whether every piece of the sphere contains at least one point that eventually (as the system evolves) returns to its original position. There are two groups (in the technical algebraic sense) that are most closely related to two dimensional dynamical systems. They are the mapping class group and the outer automorphism group. To understand the geometry of a group one must understand its geodesics; i.e. what the shortest paths are between any two points. The geodesics of the mapping class group have been well understood for some time. The principal investigator will generalize from what is known about the geodesics of the mapping class group to identify and study geodesics for the outer automorphism group.