Mathematical Sciences: Outer Automorphisms and Surface Dynamics

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9504912 Handel The project concerns the outer automorphism group of a free group and pseudo-Anosov surface dynamics. One part is a continuation of joint work with Mladen Bestvina and Mark Feighn concerning three problems. The first is a characterization of subgroups of polynomial growth outer automorphisms. The second is to extend their proof of the Tits Alternative by showing that all virtually solvable subgroups of the outer automorphism group are virtually abelian. The third is to show that the mapping torus of an outer automorphism is coherent. The goal of the other part of the project is to use combinatorial group theory methods, especially folding, to analyze a dynamically defined partial order on conjugacy classes in the mapping class group of a punctured disk. Two fields of mathematics interact in this project: dynamical systems and combinatorial group theory, the latter, perhaps surprisingly, contributing a powerful method for studying the former. In dynamical systems, the focus is on the long-term patterns that occur in a changing, or evolving system, for example a system of orbiting planets and their sun. Combinatorial group theory is the study of algebraic objects called groups using topological and combinatorial methods. One part of the project examines an important family of two-dimensional dynamical systems known as pseudo-Anosov homeomorphisms and uses a tool from combinatorial group theory called "folding" to analyze the relationship between various members of this family. Another part of the project relates to a group whose elements are the algebraic analogs of the pseudo-Anosov homeomorphisms. While most research in this area is derived from geometry, in this project the principal investigator will add dynamical systems techniques in order to investigate this group in greater detail. ***