Rigidity Aspects of Ergodic Actions of Lattices in Semisimple Lie Groups

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Abstract Katok/Furman Lattices in (higher rank) semisimple Lie groups have many remarkable properties, among them strong rigidity (Mostow) and superrigidity (Margulis, Corlette), which describe linear representations of these discrete groups in terms of the linear representations of the ambient Lie groups. One of the main themes of the proposal is to understand possible generalizations of strong rigidity and superrigidity to the framework of representations taking values in general locally compact (rather than linear, or Lie) groups. Another (but as it turns out to be a closely related) line of research is the study of the orbit structure of finite measure-preserving group actions. It turns out that ergodic finite measure-preserving actions of higher rank lattices have such a rigid orbit structure, that the group (i.e. the lattice) and the action can be reconstructed just from the produced orbit relation. Further questions and problems arise in the analysis of these orbitally rigid actions. Some progress has already been obtained in several of the proposed problems. The proposed research addresses various questions which join two disciplines: ergodic theory and Lie groups, or more generally: probability and algebra. This joining, namely the use of ergodic-theoretic methods in Lie groups and Lie groups examples in ergodic theory, has already proved itself as extremely fruitful in both fields, by solving problems and establishing new phenomena, which appeared naturally within the scope of one filed, but solutions of which required techniques and ideas from another one. The proposed research focuses on, so called, rigidity aspects of certain structures (both in ergodic theory and in Lie groups). Here rigidity means that the object, carrying certain structure, cannot be changed without a substantial change in this structure. This in particular means that the object itself can be reconstructed (or the object is determined) by this structure. Many rem arkable rigidity results are known and widely used for lattices in Lie groups, and the proposal addresses several natural generalizations of these results.