Foliations formed by generic coadjoint orbits of a class of 7-dimensional solvable Lie groups

Abstract We consider all connected and simply connected 7-dimensional Lie groups whose Lie algebras have nilradical g5,2 = span{X1, X2, X3, X4, X5 : [X1, X2] = X4, [X1, X3] = X5} of Dixmier. First, we give a geometric description 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.