#### Volume 20, issue 6 (2020)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Small $C^1$ actions of semidirect products on compact manifolds

### Christian Bonatti, Sang-hyun Kim, Thomas Koberda and Michele Triestino

Algebraic & Geometric Topology 20 (2020) 3183–3203
DOI: 10.2140/agt.2020.20.3183
##### Abstract

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 ${H}^{1}\left(S,ℝ\right)$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathsc{𝒰}$ of the trivial action in the space of ${C}^{1}$ actions of ${\pi }_{1}\left(T\right)$ on $M$ such that any action in $\mathsc{𝒰}$ 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 ${H}^{1}\left(H,ℝ\right)\ne 0$ 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.

##### Keywords
groups acting on manifolds, hyperbolic dynamics, fibered $3$–manifold, $C^1$–close to the identity
##### Mathematical Subject Classification 2010
Primary: 37C85, 57M60
Secondary: 20E22, 37D30, 57M50, 57R35
##### Publication
Revised: 17 February 2020
Accepted: 7 March 2020
Published: 8 December 2020
##### Authors
 Christian Bonatti Institut de Mathematiques de Bourgogne Universite de Bourgogne-Franche-Comté (IMB, UMR CNRS 5584) Dijon France http://bonatti.perso.math.cnrs.fr Sang-hyun Kim School of Mathematics Korea Institute for Advanced Study (KIAS) Seoul South Korea http://cayley.kr Thomas Koberda Department of Mathematics University of Virginia Charlottesville, VA United States http://faculty.virginia.edu/Koberda Michele Triestino Institut de Mathématiques de Bourgogne Universite de Bourgogne-Franche-Comté (IMB, UMR CNRS 5584) Dijon France http://mtriestino.perso.math.cnrs.fr