On cocycles with values in group extensions. Generic results

Let $1\to A\to E\to G\to 1$ be a short exact sequence of locally compact groups, $A$ amenable. Given a recurrent $G$-valued cocycle $\omega$ of an ergodic nonsingular transformation, we consider the subset of those $E$-valued cocycles whose $G$-projection is $\omega$. It is proved that for a generic cocycle from this subset the restriction of the associated (Mackey) $E$-action to $A$ is trivial. This improves some results of K. Dajani (1991, 1993) and answers a question from her paper in Trans. Amer. Math. Soc. (1991).