Non-Positive Curvature, Decision Processes and Subgroups
by
Martin Bridson
Imperial College London

How can you tell if two manifolds are "the same"; or two knots, two groups, etc.? And what horrors can lurk amongst the subgroups of seemingly benign groups?

I'll remind you how the serious study of such questions began with Dehn's work on low-dimensional manifolds. I'll then discuss recent results concerning the interplay of curvature and decision processes. In particular, I'll explain why the isomorphism problem is unsolvable in the class of combable groups (in contrast to the case of hyperbolic groups). I shall also explain why the isomorphism problem can be unsolvable among the finitely presented subgroups of a fixed product of hyperbolic groups.

This last result illustrates the fact that the subgroup structure of the fundamental groups of compact non-positively curved spaces can be rather wild. In stark contrast, we have: If H is a subgroup of a direct product of m surface groups, and if H has finitely generated homology up to dimension m, then H contains a subgroup of finite index that is itself a direct product of surface groups. I shall explain this and similar recent results.

Date received: March 4, 2002