Cocycle Rigidity and Splitting for some Discrete Parabolic Actions

TL;DR

This paper establishes trivialization of first cohomology and a splitting construction for certain discrete parabolic actions on a specific homogeneous space, advancing understanding of their cohomological properties.

Contribution

It proves the trivialization of first cohomology and introduces a splitting construction for $ ext{Z}^2$ parabolic actions on a homogeneous space, extending previous cohomological results.

Findings

First cohomology with smooth vector fields is trivial for the specified actions.

A splitting construction involving coboundary operators is developed.

Results apply to irreducible, co-compact lattices in $SL(2,\mathbb R)\times SL(2,\mathbb R)$.

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.