Small $C^1$ actions of semidirect products on compact manifolds

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 ${\psi }^{\ast }$ 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 $H^1(H,\mathbb{R})\mathrm{\ne }0$ 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.