Let be a compact fibered –manifold, presented as a mapping torus of a compact, orientable surface with monodromy , and let be a compact Riemannian manifold. Our main result is that if the induced action on has no eigenvalues on the unit circle, then there exists a neighborhood of the trivial action in the space of actions of on such that any action in is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group provided that the conjugation action of the cyclic group on 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.
"Small $C^1$ actions of semidirect products on compact manifolds." Algebr. Geom. Topol. 20 (6) 3183 - 3203, 2020. https://doi.org/10.2140/agt.2020.20.3183