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

Abstract

In this paper, we consider codimension one Anosov actions of R 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 R 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 .

Contact foliations and generalised Weinstein conjectures

“…It was conceived having in mind the problem of classification of Anosov actions of R q and the Verjovsky Conjecture for actions (c.f. [8]). The idea was to define a structure that could replicate the proprieties of Anosov contact flows in higher dimensions.…”

Contact foliations and generalised Weinstein conjectures

“…It was conceived having in mind the problem of classification of Anosov actions of R k and the Verjovsky Conjecture for actions (c.f. [7]). The idea was to define a structure which could replicate in higher dimensions the proprieties of Anosov contact flows.…”