Professors Culler and Shalen will continue their program for estimating Margulis constants, injectivity radii, and volumes of hyperbolic manifolds by using Sullivan-Patterson measures associated to 2-generator Kleinian groups. They will investigate group actions on generalized trees, extending research of Culler- Vogtmann and Gillet-Shalen. They will also apply the theory of character varieties to the study of Heegaard splittings of non- Haken 3-manifolds. Culler will continue his work on cohomological properties of the outer automorphism group of the free group. Shalen will study questions about amalgamated free product decompositions and deficiencies of subgroups, both in 3-manifold groups and in more general finitely presented groups. He will also investigate possible generalizations of Casson's invariant and connections with SL(2,C) representations of 3-manifold groups. These are all very active topics in the topology of 3- dimensional manifolds, i.e. very natural geometric objects of the same dimension as the space in which we live.
