Karen Vogtmann
Cornell University
AWM Emmy Noether Lecture
January 2007
New Orleans, LA
Abstract. Outer space was introduced in the mid1980s as a tool for studying the group Out(F_{n}) of outer automorphisms of a finitelygenerated free group. The basic philosophy is that one should think of an automorphism of a free group as a topological object, either as a homotopy equivalence of a finite graph or as a diffeomorphism of a suitable threemanifold with free fundamental group. There are compelling analogies between the action of Out(F_{n}) on Outer space and the action of an arithmetic group on a homogeneous space or the action of the mapping class group of a surface on the associated Teichmuller space. In this talk I will first describe Outer Space and explain how it is used to obtain algebraic information about Out(F_{n}). I will then indicate how Outer Space is related to other areas, from infinitedimensional Lie algebras to the mathematics of phylogenetic trees, and how ideas from Outer Space are currently expanding in new directions.
