  Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a
compact, orientable surface $S$ with monodromy $\psi$, and let $M$ be a compact
Riemannian manifold. Our main result is that if the induced action $\psi^*$ on
$H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a
neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions
of $\pi_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will
prove that the same result holds in the generality of an infinite cyclic
extension of an arbitrary finitely generated group $H$, provided that the
conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no
eigenvalues of modulus one. We thus generalize a result of A. McCarthy, which
addressed the case of abelian--by--cyclic groups acting on compact manifolds.
