Professor Shalen will continue his work with H. Gillet on the structure of groups acting on lambda-trees. They will then explore applications to hyperbolic geometry, K-theory and linear groups. Shalen will also continue his work with P. Wagreich on growth functions of groups and applications to the study of volume and diameters of hyperbolic manifolds, as well as his work with G. Baumslag and J. Morgan in which properties of finitely presented groups are derived from studying their varieties of linear representations over fields. Professors Culler and Shalen will continue their work on Dehn surgery on knots with the ultimate goal of classifying surgeries that give manifolds with cyclic fundamental groups and settling the Property P conjecture. They will also work on extending the Cyclic Surgery Theorem to links, and interpreting the new polynomial invariants for knots and links in terms of varieties of groups and representations. Professor Culler will continue his work with K. Vogtmann on the outer automorphism group of a free group. They will develop further the analogy between this group and arithmetic groups. Culler will also study the dynamics of these outer automorphisms in analogy with Thurston's work on dynamics of surface automorphisms. In addition, he will work on constructing generalized buildings for a class of goups that includes these outer antomorphism groups as well as arithmetic groups and mapping class groups. Exploring these numerous strong connections between low- dimensional topology and group theory will advance both fields, with ultimate impact on mathematical physics and other heavy users of the machinery of modern mathematics, particularly on users of precise and concise ways of describing the symmetries of highly connected spaces.
