Topological and Geometric Rigidity (07w5094)

Arriving in Banff, Alberta Sunday, July 29 and departing Friday August 3, 2007


(Indiana University)

(University of Chicago)


The rigidity phenomena associated with the names of Mostow, Margulis, Borel, Baum-Connes, and Novikov are central to differential geometry, Lie theory, geometric topology, operator index theory, and geometric group theory. While the general shape of connections between these fields are familiar to many, there are both analogies between these areas as well as idiosyncratic differences between them. This meeting will promote interaction between three communities of mathematicians: geometric topologists, geometric group theorists, and operator index analysts.

There has been significant recent progress in all three fields. This meeting aims to acquaint the members of the various communities of recent progress on topological rigidity (Farrell-Jones isomorphism conjectures, definition and calculation of UNil, and the role of the gap hypothesis in equivariant rigidity), on quasi-isometry classification of nilpotent groups using homological methods, and modern improvements on the a-T-amenability/embedding approach to Baum-Connes/Novikov conjectures.

There are a number of areas where the tools of one field seem tantalizingly close to being directly applicable to the others. As examples we mention the development of infinite dimensional technology in geometric topology (taken from the operator theorists), more refined geometric group theory being a possible avenue towards Borel conjecture results, broader application of homological methods in global analysis and algebraic simplification-generalization of tools from abstract representation theory, and the common interest in understanding the interrelations between various quasi-isometric embedding problems and the structure of boundaries of groups.

What can the various communities gain from each other? As mentioned above, all three communities have a common interest in boundary behavior of infinite discrete groups as well as embedding questions. Geometric topologists can hope to import infinite dimensional techniques from the analysts with many anticipated applications. The geometric group theorists can give the topologists a source of examples, and greater sophistication with boundary behavior and coarse geometry. The geometric group theorists can attack the boundary questions coming from geometric topology, and search for new compactifications of familiar groups that are more suitable to this circle of problems. The C*-analysts give group theorists analytic invariants, and new properties to study. The analysts, in turn, benefit from examples and large scale geometric sophistication from the geometric group theorists and applications from the geometric topologists. In short, each area stands to benefit both from techniques and problems from the others; the main goal of the conference is to introduce the practitioners of these separate but related fields to the deeper ideas of the others.