Volume 20, issue 6 (2020)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
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

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.

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
Received: 2 December 2019
Revised: 17 February 2020
Accepted: 7 March 2020
Published: 8 December 2020
Christian Bonatti
Institut de Mathematiques de Bourgogne
Universite de Bourgogne-Franche-Comté (IMB, UMR CNRS 5584)
Sang-hyun Kim
School of Mathematics
Korea Institute for Advanced Study (KIAS)
South Korea
Thomas Koberda
Department of Mathematics
University of Virginia
Charlottesville, VA
United States
Michele Triestino
Institut de Mathématiques de Bourgogne
Universite de Bourgogne-Franche-Comté (IMB, UMR CNRS 5584)