Geometric Group Theory and Surface Dynamics

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

Investigators

Abstract

The proposal divides into three projects in the related fields of two dimensional dynamical systems and geometric group theory. The first is a collaboration with John Franks to study smooth group actions on surfaces. One goal is to understand when an action by an abelian group must have a global fixed point. Another is to prove that certain lattices have essentially no non-trivial actions on the two dimensional sphere. The second project is joint work with Benson Farb. One part of this project is to study lifts of the mapping class group of a surface to the diffeomorphism group of that surface. The third project continues joint work with Lee Mosher. The action of the mapping class group on Teichmuller space is paralleled by the action of the outer automorphism group on Outer Space. The goal of this collaboration is to formulate an analogue in Outer Space of Teichmuller geodesics. 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 is its very close connection to mapping class groups. One part of the proposal focuses on finding global fixed points of actions; i.e. points that are stationary for every diffeomorphism associated to the group. Another is to understand actions of the mapping class group of a surface on that surface. 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. A third goal of the project is to generalize from what is known about the geodesics of the mapping class group to identify and study geodesics for the outer automorphism group.