On integrable codimension one Anosov actions of $\RR^k$

Thierry Barbot; Carlos Maquera

doi:10.3934/dcds.2011.29.803

Article Contents

2011, Volume 29, Issue 3: 803-822. Doi: 10.3934/dcds.2011.29.803

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On integrable codimension one Anosov actions of $\RR^k$

Thierry Barbot^1, and
Carlos Maquera^2,

1.
Université d'Avignon et des pays de Vaucluse, LANLG, Faculté des Sciences, 33 rue Louis Pasteur, 84000 Avignon
2.
Universidade de São Paulo - São Carlos, Instituto de Ciências Matemáticas e de Computação, Av. do Trabalhador São-Carlense 400, 13560-970 São Carlos, SP

Received: December 31, 2009

Revised: May 31, 2010

Published: August 2011

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Abstract

In this paper, we consider codimension one Anosov actions of $\RR^k,\ k\geq 1,$ on closed connected orientable manifolds of dimension $n+k$ with $n\geq 3$. We show that the fundamental group of the ambient manifold is solvable if and only if the weak foliation of codimension one is transversely affine. We also study the situation where one $1$-parameter subgroup of $\RR^k$ admits a cross-section, and compare this to the case where the whole action is transverse to a fibration over a manifold of dimension $n$. As a byproduct, generalizing a Theorem by Ghys in the case $k=1$, we show that, under some assumptions about the smoothness of the sub-bundle $E^{ss}\oplus E$^uu , and in the case where the action preserves the volume, it is topologically equivalent to a suspension of a linear Anosov action of $\mathbb{Z}^k$ on $\TT^{n}$.

Keywords:

Anosov action,
Verjovsky conjecture.

Mathematics Subject Classification: Primary: 37C85.

Citation:

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