Mathematical Sciences: Geometry and Low-Dimensional Topology in Group Theory

9703756 Sela Zlil Sela intends to complete his sequence of papers on the isomorphism problem for hyperbolic groups and the sequence of papers on automorphisms of free groups. He further plans to complete writing his work on the structure of sets of solutions to systems of equations in a free group, and on families of such systems depending on a set of defining parameters. This work is based on borrowing notions and ideas from algebraic geometry and low dimensional topology and implementing them in group theory. Sela and E. Rips (Hebrew University) are engaged in a long-standing investigation of fundamental algebraic structures known as ``groups,'' structures which abstract and capture the essence of symmetry. One of three key problem of the field is the isomorphism problem, the problem of telling when two groups, presented differently, are really the same. It is known that there can be no general recipe for solving this problem, one that will work for a completely arbitrary pair of groups. For this reason, recipes have been sought that work within large classes of interesting groups, and Sela is closing in on such a recipe for the class of (Gromov) hyperbolic groups, groups whose interest derives from being attached to geometric objects known as manifolds, both arbitrary manifolds of dimension three and negatively curved ones of any dimension. The ideas and techniques evolved in this work have led to other questions and to unexpected answers, all bearing on the deep and intimate connection between the algebra of groups and the geometry of manifolds. During the last year, Sela has developed a geometric approach to sets of solutions of equations in a so-called free group. The approach borrows heavily from algebraic geometry. More recently he has pushed this geometric approach further in order to study indexed families of sets of solutions of equations in groups. Sela and Rips now expect that the structure theory they have developed will find many applications in group theory and in mathematical logic. In particular, they believe that the tools they have in hand will provide positive answers to several questions posed by the logician Alfred Tarski around 1950. Since their techniques borrow heavily from low dimensional topology, they hope that their structure theory will also find applications to basic questions in low-d topology. ***