Small C1 actions of semidirect products on
compact manifolds
Let T be a compact fibered 3-manifold, presented as a mapping torus of a compact, ori-entable surface S with monodromy ψ, and let M be a compact Riemannian manifold. Our main result is that if the induced action ψ˚on H 1 pS, Rq has no eigenvalues on the unit circle, then there exists a neighborhood U of the trivial action in the space of C 1 actions of π 1 pT q on M such that any action in 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 pH, Rq ‰ 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.