Geometric Group Theory and Surface Dynamics

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AbstractAward: DMS-0103435Principal Investigator: Michael HandelThe proposal divides into four projects in the related fields oftwo 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 automorphismgroup. The second is to continue the principal investigator'sstudy of the forcing order on braid types in the four timespunctured disk. This project has both an experimental andtheoretical part. Computer programs are used to generateexamples and to aid in the formation of conjectures. Onceconjectures are made that cannot be disproved by computer,rigorous proofs will be attempted. The third project is acollaboration with John Franks with the long term goal of provingthat generic area preserving diffeomorphisms of the twodimensional sphere have dense periodic orbits. The moreimmediate goal is to find analogs for generic area preservingdiffeomorphisms of the topological structure that is known toexist for twist maps. The fourth is a collaboration with LeeMosher. The first steps in the project will be to identify andstudy quasi-lines that can play the role in Culler Vogtmann spacethat Teichmuller geodesics play in Teichmuller space.This proposal is concerned with the interface between two areasof mathematics: two dimensional dynamical systems and geometricgroup theory. The former studies the long term behavior ofsystems that evolve over time, while the latter treats algebraicobjects by geometric means. The fields have been intertwined formore than fifty years and have been the focus of a great deal ofresearch in the past twenty. Part of the proposal focuses on twolong standing fundamental questions in two dimensional dynamicalsystems. The first examines how simple systems change intochaotic ones. The second concerns transformations of the spherethat preserve area, and asks whether every piece of the spherecontains at least one point that eventually (as the systemevolves) returns to its original position. There are two groups(in the technical algebraic sense) that are most closely relatedto two dimensional dynamical systems. They are the mapping classgroup and the outer automorphism group. To understand thegeometry of a group one must understand its geodesics; i.e. whatthe shortest paths are between any two points. The geodesics ofthe mapping class group have been well understood for some time.The principal investigator will generalize from what is knownabout the geodesics of the mapping class group to identify andstudy geodesics for the outer automorphism group.