Polycyclic groups and transversely affine foliations
A foliation is called transversely affine if coordinates can be chosen so that the holonomy maps are all affine. It is known that all polycyclic groups occur as the fundamental group of a certain affinely flat manifold. It is conjectured that the converse is virtually true [see \textit{J. Milnor}, Adv. Math. 25, 178-187 (1977; Zbl 0364.55001)]. The author considers here codimension one foliations of manifold with polycyclic fundamental group and shows that in certain natural situations the foliation must have a transverse affine structure. For example, the following theorem is proved. Theorem: Suppose \(\mathcal F\) is a real analytic foliation which has a compact leaf space. If \(\pi_ 1(M)\) is virtually polycyclic and \(\mathcal F\) does not have any invariant measures, then \(\mathcal F\) is minimal and transversely affine. If in addition, \(M\) is compact, then \(\mathcal F\) is covered by a product foliation.
scientific article; zbMATH DE number 120211