Volume Preserving Codimension One Anosov Flows in Dimensions Greater than Three Are Suspensions

Verjovsky stated the conjecture in [35] for all dimensions with an additional assumption that the fundamental group of the manifold is solvable. This was proved by Plante [24, 25] and Armendariz [4], who showed that the conjecture is true if and only if the fundamental group of the manifold is solvable. In the above form, the conjecture first appeared in Ghys [13]. However, Ghys has pointed out that Verjovsky had originally proposed it in the 1970’s. In [13], Ghys showed that the conjecture is true if the sum E of the strong bundles of the flow is of class C1 or if the codimension one center stable bundle E is C2 and the flow preserves volume. The first result of Ghys was generalized in [30] to Lipschitz E. The second one was extended in [31] to the case when E is Lip− or when E is C1+Lip−, where Lip– means C for all θ ∈ (0, 1). Guelman [14] showed that the conjecture is true in dimension four if the time one map of the flow can be C1 approximated by an Axiom A diffeomorphism. Her method works also in dimension three. In this paper, we prove the following result.