Local rigidity of homogeneous parabolic actions: I. A model case

We show a weak form of local differentiable rigidity for the rank 2 abelian action of upper unipotents on SL(2,R) × SL(2,R) Γ. Namely, for a 2- parameter family of sufficiently small perturbations of the action, satisfying certain transversality conditions, there exists a parameter for which the perturbation is smoothly conjugate to the action up to an automorphism of the acting group. This weak form of rigidity for the parabolic action in question is optimal since the action lives in a family of dynamically different actions. The method of proof is based on a KAM-type iteration and we discuss in the paper several other potential applications of our approach.