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2020 Small $C^1$ actions of semidirect products on compact manifolds
Christian Bonatti, Sang-hyun Kim, Thomas Koberda, Michele Triestino
Algebr. Geom. Topol. 20(6): 3183-3203 (2020). 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.

Citation

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Christian Bonatti. Sang-hyun Kim. Thomas Koberda. Michele Triestino. "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

Information

Received: 2 December 2019; Revised: 17 February 2020; Accepted: 7 March 2020; Published: 2020
First available in Project Euclid: 16 December 2020

MathSciNet: MR4185939
Digital Object Identifier: 10.2140/agt.2020.20.3183

Subjects:
Primary: 37C85, 57M60
Secondary: 20E22, 37D30, 57M50, 57R35

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.20 • No. 6 • 2020
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