Geometry and dynamics of the outer automorphism group of a free group

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A successful program for studying the outer automorphism group Out(F) of a free group has been based on its relationship to the mapping class group Mod(S) of a surface. Both groups play an important role in many areas of mathematics, including algebraic geometry, geometric group theory, and low-dimension topology; they have also entered into some recent developments in theoretical physics. They are thought to be generalizations of, and indeed share many properties with, linear groups. In turn linear groups, i.e., groups consisting of matrices, have been well studied and appear everywhere in mathematics, physics, and computer science. Because matrices are relatively easy to understand, it is productive when studying a particular group to understand how closely it resembles a linear group. My research is part of this larger program of understanding possible obstructions to Out(F) and Mod(S) being linear groups, as well as understanding the extent to which their behaviour nonetheless resembles that of linear groups. The methods I use are those of geometric group theory, working on the principal that the algebraic properties of groups are reflected in the geometry and dynamics of the spaces they act on.