Mathematical Sciences: Geometric Group Theory and 3-Manifolds

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9503034 Stallings The project is to study hyperbolic groups, automatic groups, non-positively curved triangles of groups, and related matters involving geometric questions about group-theory. The problem of distinguishing non-hyperbolic automatic groups has several interesting aspects. There is a notion of angle between pairs of subgroups A,B of a group G, which involves the smallest length of an expression aba'b'a"b"... which is reduced and gives the identity element of G. This leads to interesting algorithmic questions; in the case of free groups this relates to the Loop Theorem. Another algorithmic question about "inner rank" is related to questions about surfaces in 3-manifolds. Groups arise in all branches of mathematics, as sets of symmetries, i.e. transformations which preserve structure. The investigator hopes to understand various properties of groups, the classification of types of groups, and the use of group theory to connect mathematical areas. In geometry and topology, groups naturally arise as subgroups of matrix groups and the fundamental groups of spaces. The fundamental group as defined by Poincare is the basic ingredient in the classification of three-dimensional manifolds, and major problems are group-theoretic. This is especially true in dimension three. There is also an intrinsic large-scale geometry inherent in an abstract group, which has led to the recent interest in what are known as hyperbolic groups. This leads in turn to algorithmic problems and to formal language theory and will some day be of interest and use to computer scientists. The project has identified some problem areas as described above, "angle," "inner rank," and so on, which are of interest in the interaction of algebra and geometry; some progress in these areas has been made; they are becoming active areas of research, in particular, by the investigator and his Ph.D. students. ***