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Ergodic Group Theory

Final Report Summary - UB12 (Ergodic Group Theory)

The main achievement of the project was the development, together with A. Furman, of the theory of Algebraic Representations of Ergodic Actions.
Studying equivariant maps between ergodic spaces to algebraic varieties from a categorical perspective we found new phenomena and provided new tools for understanding ergodic actions of algebraic groups.
Using this theory we ended up proving the results we aimed at while proposing this project.
In particular, we proved various Super-Rigidity theorems for lattices (and also non-lattices) in locally compact groups.
In our super-rigidity theorems, the target groups are algebraic groups over arbitrary complete valued fields, as opposed to merely local fields as in the classical case.
Our proof also applies to lattices in a wide class of groups, strictly containing the classical class of higher rank semisimple groups.

The work alluded to in the above is based on two supplementary works, one was carried with Tsachik Gelander and one Bruno Duchesne and Jean Lecureux.
With Gelander we wrote the paper "Equicontinuous actions of semisimple groups" (submitted for publication), in which we study equicontinuous actions of semisimple groups and some generalizations. We prove that any such action is universally closed, and in particular proper. We give various applications including closedness of continuous homomorphisms, metric ergodicity of transitive actions and vanishing of matrix coefficients for reflexive (more generally: WAP) representations.
With Duchesne and Lecureux we wrote the paper "Almost algebraic actions of algebraic groups and applications to algebraic representations" (submitted for publication), in which we consieder an algebraic G group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis-Zimmer super-rigidity phenomenon.

Another major achievement we had, again together with Alex Furman, is our new development of a method that uses boundary theory in order to achieve results regarding simplicity of the Lyapunov spectrum. Our main results are yet not properly written, but we provided a summery in "Boundaries, rigidity of representations, and Lyapunov exponents" (to appear at the proceedings of the ICM 2014). In this paper we discussed some connections between measurable dynamics and rigidity aspects of group representations and group actions. A new ergodic feature of familiar group boundaries is introduced, and is used to obtain rigidity results for group representations and to prove simplicity of Lyapunov exponents for some dynamical systems.