9803461 Swenson First, Swenson will continue his collaboration with Martin Dunwoody on the algebraic torus theorem. Much of the work done in the field of geometric group theory has been generalizing results from 3-manifold theory to the setting of finitely generated groups. The first result to generalize was the sphere theorem, which was generalized by Stallings to the Ends Theorem. When the most general versions of the annulus and torus theorem for three-manifolds were proven by Scott in 1980, he conjectured that there should be a purely algebraic analog to these theorems. The first step in that direction was a proof of the algebraic annulus theorem for word hyperbolic groups by Scott and Swarup in 1995. The algebraic annulus theorem for finitely generated groups was proven by Dunwoody and Swenson in 1996. These two will extend this result to an n-dimensional algebraic torus theorem for finitely generated groups. In addition, Swenson will continue his investigations into the boundaries of groups. The study of group boundaries was begun in the theory of Kleinian and Fuchsian groups. The first person to apply these ideas to abstract groups was Gromov. He defined the visual boundary both of a word-hyperbolic group, and of a CAT(0) group. It was shown by Bestvina and Mess that these boundaries carry considerable cohomological information about the group. Swenson has two different projects in this area. First, he wishes to determine the relationship between the local and global homology in the visual boundary of a word-hyperbolic group. Bestvina began the work in this area, after which Swenson made a contribution. There are, however, certain important questions that remain open. The second project concerns the existence (or non-existence) of cut points in these boundaries. It has been shown by Bowditch and Swarup that the visual boundary of a word-hyperbolic group cannot contain a cut-point. Swenson will try to "extend" this result to the nonpositive ly curved or CAT(0) setting. He will also look for consequences of boundaries' not having cut-points (semistability at infinity, for example). Years ago it was said that infinite group theory consisted entirely of counter-examples. Geometric group theory has done much to remedy this situation. Thurston's work towards the classification of all 3-manifolds, spaces which look locally like 3-space, was the origin of geometric group theory. Z. He, Cannon, and others noticed the amazing correlation between the group theoretic properties of the fundamental group of a geometric 3-manifold and the geometric properties of its universal cover. Much of the work done in geometric group theory has been to translate techniques and sometimes even theorems from the setting of 3-manifolds to the seemingly much more general setting of simply connected 2-complexes, spaces formed by gluing triangles along their edges (often with more than two triangles sharing a common edge). The torus theorem for 3-manifolds says that any time there is an essential map of a 2-torus (an inner tube) into a 3-manifold M, either M is very simple and well understood, or there is an embedded 2-torus along which M can be cut to yield simpler pieces. Swenson is working with Martin Dunwoody to extend this result to apply to any essential map of a torus of any dimension into a simply connected 2-complex. In the second part of the project, Swenson will investigate non-positively curved 2-complexes, using techniques transferred over from differential geometry by Gromov and others. ***
