Mathematical Sciences: Real Trees and Free Groups

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The mapping class group of a closed surface acts on compactified Teichmuller space. The topology of this space, as well as the dynamics of this action, are well understood and give information about global properties of the mapping class group. For example, it satisfies the Tits alternative. The project is to continue an analogous study of the outer automorphism group of the free group via its action on compactified outer space. Points in Outer Space are equivalence classes of simplicial metric trees. The compactifying points are classes of real trees, objects obtained from degenerations of simplicial trees. Groups abound in the physical sciences because they arise as the set of symmetries of an object. There is a special group, called the free group, from which all others may be constructed. The project is to continue the investigation of the basic structure of the free group. The technique is to use tools recently developed by E. Rips and generalized by the investigator with M. Bestvina.