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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 H1(S, ) has no eigenvalues on the unit circle, then there exists a neighborhood 𝒰 of the trivial action in the space of C1 actions of π1(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 H1(H, )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
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