Rigidity Phenomena of Group Actions on a Class of Nilmanifolds and Anosov R^n Actions

Abstract

An action of a group Г on a manifold M is a homomorphism ρ from Г to Diff(M). ρo is locally rigid if the nearby homomorphism ρ, ρ(γ) = h o ρ0, (γ) 0h^(-1) for some h Є Diff(M) and for all, γ Є Г. In other words, ρ0 is isolated from other actions up to a smooth conjugation. In this thesis we studied some standard group actions on a broader class of manifolds, the free, k-step nilmanifolds N(n, k); we obtained that the standard SL(n, Z) action on N(n,2) is locally rigid for n = 3, and n ≥ 5. We recall that N(n,1) = T^n. Hence, our results are the generalization to the local rigidity result for the standard action on torus T^n. We observed also, for the first time, that for discrete subgroups Aut(n, 2) of a Lie group, which is not even reductive, the action on N(n,2) is deformation-rigid for n = 3, and n ≥ 5. We also investigated the dynamics of Anosov R^n actions and obtained a number of results parallel to those of Anosov diffeomorphisms and flows. E.g., the strong stable (unstable) manifold for a regular element is dense iff the action is weakly mixing (for a volume-preserving action); an Anosov action with no dense, strong stable (unstable) manifold can always be reduced to suspension of the action mentioned above; there are two compatible measures to the Anosov actions.