Cohomological Approach to Rigidity in Geometry and Dynamics

University Of Chicago, Chicago IL

Investigators

Abstract

The PI plans to establish rigidity and finiteness results for infinite groups, both in geometric and dynamical contexts. In an attempt to broaden the scope of rigidity as initiated by Mostow, Margulis, Zimmer and others, we aim at results for very general (finitely generated) groups, notably fundamental groups arising in geometry. The proposed methods are however also new and relevant for the more classical study of arithmetic and S-arithmetic groups. There are three aspects to the project. The first is to construct cohomological invariants characterizing various geometric situations; a typical example is the interpretation of negative curvature in terms of bounded cohomology that we propose with Shalom. The second step is to develop a suitable theory to handle these invariants in an efficient way, notably by using tools from ergodic theory. Last, one has to apply this to concrete situations. We focus in particular on superrigidity and cocycle superrigidity for non-linear groups and on orbit equivalence of ergodic actions. The last forty years have witnessed the birth and remarkable development of what is now called ''rigidity theory''. The first discovery in that field was this: Even though a flat space (which is the kind of space one thought we live in, from the ancient Greeks to Newton) can be deformed in many ways, some curved spaces (whose relevance to our world has been discovered by Einstein) are in a sense rigid. Later came a discovery regarded as yet much more fundamental: The most symmetric geometric spaces turn out to be in fact arithmetic objects. The project under consideration here is to broaden the scope this rigidity theory to many more geometric spaces, and especially to deal with situations in which the geometry cannot be reduced to arithmetics. Actually, we propose new methods that stand out by their abstract nature and allow us to tackle situations which do not even stem from geometry, but rather regard the study of dynamics. (Dynamics is the theory of systems that evolve in time, typically ocean currents or galaxies, and may have chaotic behaviours.) Our project proposes new methods for obtaining more results both in this dynamical setting and in geometry, and this in a unified way.