A fundamental technique for studying a group G is to view G as a group of automorphisms of a geometric object X. Geometric and topological properties of X can then be used to study algebraic properties of G. Beautiful classical examples of this are the theory of arithmetic and S-arithmetic groups acting on homogeneous spaces and buildings, including work of Borel and Serre on cohomological properties of these classes of groups, and the theory of groups of surface homeomorphisms acting on the Teichmüller space of the surface. Professor Karen is interested in developing geometric theories for other classes of groups. In particular, she has worked with orthogonal and symplectic groups, SL(2) of rings of imaginary quadratic integers, groups of automorphisms of free groups, and mapping class groups of surfaces. Her main focus in recent years has been on the group of outer automorphisms of a free group, where the appropriate geometric object is called Outer space. This space turns out to have surprising connections with certain infinite-dimensional Lie algebras (discovered by Kontsevich) and also with the study of phylogenetic trees in biology.