Codimension one foliations on solvable manifolds (Q1312817)

Throughout the paper \(M\) is a smooth connected manifold with solvable fundamental group \(\Pi\), \({\mathcal F}\) is a transversely oriented codimension one \(\text{C}^ r\) \((r\geq 0)\) foliation without closed leaves and such that every nontrivial leaf holonomy has an isolated fixed point, and \(\widetilde {\mathcal F}\) is the lift of \({\mathcal F}\) to the universal covering space \(\widetilde {M}\) of \(M\). The results read as follows: (1) There exists a \(\text{C}^ r\) submersion \(D\widetilde {M}\to \mathbb{R}\) constant along the leaves of \(\widetilde {F}\) and a homomorphism \(\varphi: \Pi\to \text{Diff}^ r (\mathbb{R})\) such that \(D(\gamma x)= \varphi (\gamma) D(x)\) for all \(x\in \widetilde {M}\) and \(\gamma\in \Pi\). (2) If all the leaves of \({\mathcal F}\) are dense in \(M\), then \({\mathcal F}\) is topologically conjugate to a transversely affine foliation. (3) If either \(\Pi\) is polycyclic or all the leaves of \({\mathcal F}\) have finitely generated fundamental groups, then \({\mathcal F}\) is complete, i.e. the leaf space \(\widetilde {M}/ \widetilde{\mathcal F}\) of \(\widetilde {\mathcal F}\) is homeomorphic to \(\mathbb{R}\).