GERSTEN, 98-00158 Gersten plans to continue his work on homological invariants in geometric group theory. He has given both homological and cohomological characterizations of hyperbolic groups and a homological characterization of distortion of a subgroup of a group. He plans to use these invariants to study open questions concerning Gromov's bounded cohomology, the word problem for finitely presented groups, and the isomorphism problem for hyperbolic groups, among others. The area of geometric group theory uses the methods of geometry to study symmetry. This turns upside down F. Klein's Erlanger Program of the 19th century, which proposed studying a geometry by means of its symmetries. For example, in Klein's program the geometry of the Euclidean plane would be studied by means of its symmetries, whose generators are rotations, reflections and parallel motions, whereas in geometric group theory, one would study which properties of the symmetries one could recover from Euclidean geometry, when one ignores small-scale features and concentrates only on what is visible to a distant observer. It is a remarkable fact that the word problem, which concerns writing a computer program to decide when two symmetries are the same when they are written in terms of generators, depends only on the large-scale geometry. The word problem is a paradigm for a large class of problems which can be studied by the methods of this area.
