Small $C^1$ actions of semidirect products on compact manifolds
| Autor: | Christian Bonatti, Thomas Koberda, Sang-hyun Kim, Michele Triestino |
|---|---|
| Přispěvatelé: | Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Centre National de la Recherche Scientifique (CNRS), School of Mathematics (KIAS Séoul), Korea Institute for Advanced Study (KIAS), University of Virginia [Charlottesville], ANR-19-CE40-0007,Gromeov,Groupes d'homéomorphismes de variétés(2019) |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
37D30 [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] Cyclic group Dynamical Systems (math.DS) Group Theory (math.GR) 01 natural sciences [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] 57M60 $C^1$–close to the identity Mathematics - Geometric Topology Primary 37C85. Secondary 20E22 57K32 [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] 0103 physical sciences Mapping torus FOS: Mathematics 57R35 20E22 0101 mathematics Abelian group Mathematics - Dynamical Systems Mathematics 37C85 010102 general mathematics Geometric Topology (math.GT) groups acting on manifolds Riemannian manifold Surface (topology) 57M50 fibered $3$–manifold hyperbolic dynamics Unit circle Monodromy 010307 mathematical physics Geometry and Topology Finitely generated group Mathematics - Group Theory |
| Zdroj: | Algebraic and Geometric Topology Algebraic and Geometric Topology, Mathematical Sciences Publishers, 2020, 20 (6), pp.3183-3203. ⟨10.2140/agt.2020.20.3183⟩ Algebr. Geom. Topol. 20, no. 6 (2020), 3183-3203 |
| ISSN: | 1472-2747 1472-2739 |
| DOI: | 10.2140/agt.2020.20.3183⟩ |
| Popis: | Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $\psi$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $\psi^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $\pi_1(T)$ on $M$ such that any action in $\mathcal{U}$ 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 $H^1(H,\mathbb{R})\neq 0$ 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. Comment: 11 pages; final version to appear in Algebraic & Geometric Topology |
| Databáze: | OpenAIRE |
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