Cocycle Rigidity and Splitting for some Discrete Parabolic Actions

Abstract

We prove trivialization of the first cohomology with coefficients in smooth vector fields, for a class of ${\mathbb{Z}}^2$ parabolic actions on $(SL(2,\mathbb{R})\times SL(2,\mathbb{R}))/\Gamma$, where the lattice $\Gamma$ is irreducible and co-compact. We also obtain a splitting construction involving first and second coboundary operators in the cohomology with coefficients in smooth vector fields.

21 pages