9705007 Meier Meier will continue determining feasible methods for computing the Sigma-invariants of infinite groups. The Sigma-invariants of a group G determine the finiteness properties of normal subgroups above the commutator subgroup of G; they are also closely connected to the Thurston norm on the homology of a 3-manifold. A program will be started to find concrete examples illuminating the recent extension (due to Bieri and Geoghegan) of the Sigma-invariants to a theory encompassing group actions on CAT(0) spaces. In another direction, Meier will study generalizations of Cannon's almost convex property that are presentation independent and that still have strong topological and computational properties. Group theory was originally introduced into the sciences through the study of symmetry: symmetry of naturally occurring objects, molecules, solutions of equations, etc. Geometric group theory proceeds by re-introducing geometric techniques into a field that had become highly algebraic. Most of Meier's work focuses on two main ideas: (1) exploring the implications of certain geometric structures in terms of algebraic properties; and, (2) looking for computationally effective means of working with important families of groups. Often a syncretic approach, combining the geometric and computational perspectives, yields the strongest results. ***
