On integrable codimension one Anosov actions of IR^k

TL;DR

This paper investigates the structure of codimension one Anosov actions of IR^k on manifolds, establishing conditions for solvability of the fundamental group and characterizing the actions as suspensions of linear Anosov actions under certain assumptions.

Contribution

It characterizes when the fundamental group is solvable and generalizes Ghys's theorem to higher dimensions, linking Anosov actions to suspensions of linear models.

Findings

Fundamental group is solvable iff the weak foliation is transversely affine.

Under certain smoothness and volume-preserving conditions, actions are topologically suspensions of linear Anosov actions.

Comparison of actions with cross-sections to fibrations over lower-dimensional manifolds.

Abstract

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.