Mathematical Sciences: Automorphisms of the Free Group and Their Application to the Dynamics of Surface Diffeomorphisms

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8904934 Handel The principal investigator will work on a pair of related projects, one concerning automorphisms of the free group and the other concerning the dynamics of surface diffeomorphisms. The first is a continuation of joint work with Mladen Bestvina. They hope to show that the group of outer automorphisms of the free group of rank n satisfies the Tits Alternative. They also hope to complete the characterization of geometric automorphisms that was begun earlier. The second project involves the computer implementation of algorithms developed earlier to determine the Thurston canonical form for an element of the mapping class group of a compact surface. Particular emphasis will be placed on the genus zero case, the goal being to understand the dynamics of a diffeomeophism f of the two-sphere by studying the canonical forms associated to f outside a finite invariant set.