We effectively conclude the local rigidity program for generic restrictions of partially hyperbolic Weyl chamber flows. Our methods replace and extend previous ones by circumventing computations made in Schur multipliers. Instead, we construct a natural topology on H 2 (G, Z), and rely on classical Lie structure theory for central extensions.

We study nontrivial entropy invariants in the class of parabolic flows on homogeneous spaces, quasi-unipotent flows. We show that topological complexity (ie, slow entropy) can be computed directly from the Jordan block structure of the adjoint representation. Moreover using uniform polynomial shearing we are able to show that the metric orbit growth (ie, slow entropy) coincides with the topological one, establishing hence variational principle for quasi-unipotent flows (this also applies to the non-compact case). Our results also apply to sequence entropy. We establish criterion for a system to have trivial topological complexity and give some examples in which the measure-theoretic and topological complexities do not coincide for uniquely ergodic systems, violating the intuition of the classical variational principle. * K.

We show local and cocycle rigidity for R k × Z l partially hyperbolic translation actions on homogeneous spaces G/Λ. We consider a large class of actions whose geometric properties are more complicated than previously treated cases. It is also the first time that partially hyperbolic twisted symmetric space examples have been treated in the literature. The main new ingredient in the proof is a combination of geometric method and the theory of central extensions.

We study Kakutani equivalence in the class of unipotent flows acting on compact quotients of semisimple Lie groups. For every such flow we compute the Kakutani invariant of M. Ratner, the value of which being explicitly given by the Jordan block structure of the unipotent element generating the flow. This, in particular, answers a question of M. Ratner. Moreover, it follows that the only standard unipotent flows are given by 1 t 0 1 × id acting on (SL(2, R) × G ′ )/Γ ′ , where Γ ′ is an irreducible lattice in SL(2, R) × G ′ (with the possibility that G ′ = {e}).

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