Cohomological Approach to Rigidity in Geometry and Dynamics

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The PI plans to establish rigidity and finiteness results forinfinite groups, both in geometric and dynamical contexts. In anattempt 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 arisingin geometry. The proposed methods are however also new andrelevant for the more classical study of arithmetic andS-arithmetic groups. There are three aspects to the project.The first is to construct cohomological invariants characterizingvarious geometric situations; a typical example is theinterpretation of negative curvature in terms of boundedcohomology that we propose with Shalom. The second step is todevelop a suitable theory to handle these invariants in anefficient way, notably by using tools from ergodic theory. Last,one has to apply this to concrete situations. We focus inparticular on superrigidity and cocycle superrigidity fornon-linear groups and on orbit equivalence of ergodic actions.The last forty years have witnessed the birth and remarkabledevelopment of what is now called ''rigidity theory''. The firstdiscovery in that field was this: Even though a flat space (whichis the kind of space one thought we live in, from the ancientGreeks to Newton) can be deformed in many ways, some curvedspaces (whose relevance to our world has been discovered byEinstein) are in a sense rigid. Later came a discovery regardedas yet much more fundamental: The most symmetric geometric spacesturn out to be in fact arithmetic objects. The project underconsideration here is to broaden the scope this rigidity theoryto many more geometric spaces, and especially to deal withsituations in which the geometry cannot be reduced toarithmetics. Actually, we propose new methods that stand out bytheir abstract nature and allow us to tackle situations which donot even stem from geometry, but rather regard the study ofdynamics. (Dynamics is the theory of systems that evolve intime, typically ocean currents or galaxies, and may have chaoticbehaviours.) Our project proposes new methods for obtaining moreresults both in this dynamical setting and in geometry, and thisin a unified way.