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Small $C^1$ actions of
semidirect products on compact manifolds
Christian Bonatti, Sang-hyun Kim, Thomas Koberda and Michele Triestino |
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Algebraic & Geometric Topology 20 (2020) 3183–3203
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DOI: 10.2140/agt.2020.20.3183
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Abstract |
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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. |
Keywords
groups acting on manifolds, hyperbolic dynamics, fibered
$3$–manifold, $C^1$–close to the identity
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Mathematical Subject Classification 2010
Primary: 37C85, 57M60
Secondary: 20E22, 37D30, 57M50, 57R35
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References |
Publication
Received: 2 December 2019
Revised: 17 February 2020
Accepted: 7 March 2020
Published: 8 December 2020
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Authors |
| Christian Bonatti | |
| Institut de Mathematiques de
Bourgogne Universite de Bourgogne-Franche-Comté (IMB, UMR CNRS 5584) Dijon France |
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| http://bonatti.perso.math.cnrs.fr | |
| Sang-hyun Kim | |
| School of Mathematics Korea Institute for Advanced Study (KIAS) Seoul South Korea |
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| http://cayley.kr | |
| Thomas Koberda | |
| Department of Mathematics University of Virginia Charlottesville, VA United States |
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| http://faculty.virginia.edu/Koberda | |
| Michele Triestino | |
| Institut de Mathématiques de
Bourgogne Universite de Bourgogne-Franche-Comté (IMB, UMR CNRS 5584) Dijon France |
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| http://mtriestino.perso.math.cnrs.fr | |
