Geometric Group Theory and Surface Dynamics

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The project concerns pseudo-Anosov surface dynamics and the outer automorphism group of a free group. One part of the project is to analyze a certain partial order on conjugacy classes in the mapping class group of a four times punctured disk. The goal is to find a description of this partial order, in terms of the known combinatorial structure of the mapping class group, that is more transparent and more amenable to computation than the standard dynamical one. The second part of the project extends the principal investigator's joint work with Mark Feighn on coherence of groups. The focus is on deciding which one-relator groups are coherent. The third and final part of the project is a continuation of work with Mladen Bestvina and Mark Feighn on the outer automorphism group of a free group. The goal of this part of the proposal is to formulate and prove a structure theorem for a class of finitely generated subgroups that contains, for example, all abelian subgroups. The pseudo-Anosov homeomorphism is one of the most important concepts in the areas of mathematics known as three-dimensional geometry, three-dimensional topology, and two-dimensional dynamical systems. The study of pseudo-Anosov homeomorphisms began in the mid-1970's. During the decade that followed, a great deal of progress was made in analyzing their behavior. Since then progress has slowed, in part because only the more difficult problems remain. Chief among these problems is an understanding of the "forcing partial order," which, roughly speaking, describes the way in which simple two dimensional systems change into chaotic ones. The principal investigator will use extensive computer calculations and techniques from other areas of mathematics, especially that of geometric group theory to investigate this problem. The application of techniques derived from a variety of mathematical subfields should prove effective in illuminating this problem. The other parts of the project focus on the algebraic analogs of the p seudo-Anosov homeomorphisms and will use techniques that are common to dynamical systems and geometric group theory.