On integrable codimension one Anosov actions of IR^k

Thierry Barbot (1) , Carlos Maquera (2)

1 LANLG - Laboratoire d'Analyse non linéaire et Géométrie
2 Instituto de Ciências Matemáticas e de Computação

In this paper, we consider codimension one Anosov actions of IR^k, k ≥ 1, on closed connected orientable manifolds of dimension n+k with n≥ 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 IR^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 ⊕ 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 Z^k on T^n.

Mots clés: Anosov action; Verjovsky conjecture

Domaines: Systèmes dynamiques [math.DS]

Thierry Barbot, Carlos Maquera. On integrable codimension one Anosov actions of IR^k. Discrete and Continuous Dynamical Systems - Series A, 2011, 29 (3), pp.803--822. ⟨hal-00450685⟩