Small C1 actions of semidirect products on compact manifolds

Bonatti, Christian; Kim, Sang-hyun; Koberda, Thomas; Triestino, Michele

doi:10.2140/agt.2020.20.3183

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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 $ψ$ , and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $ψ^{*}$ on $H^{1} (S, ℝ)$ 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 $π_{1} (T)$ 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 $H^{1} (H, ℝ) \neq 0$ 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.

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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

References

Publication

Received: 2 December 2019

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

Volume 20, issue 6 (2020)