Measurable Foliations Associated to the Coadjoint Representation of a Class of Seven-Dimensional Solvable Lie Groups

Abstract

We consider connected and simply connected seven-dimensional Lie groups whose Lie algebras have nilradical ${\frak g}_{5,2}$ of Dixmier. First, we give geometric descriptions of the maximal-dimensional orbits in the coadjoint representation of all considered Lie groups. Next, we prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. Finally, the topological classification of all these foliations is also provided.