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* on H-1 (S, R) has no eigenvalues on the unit circle, then there exists a neighborhood Li of the trivial action in the space of C-1 actions of pi(1) (T) on M such that any action in Li 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, R) not equal 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.