Lattices of a Lie group and Seifert fibrations (Q1118210)

Let L be a Lie group with finitely many components, K a maximal compact subgroup of L, and \(\Lambda\) a lattice of L. \(\Lambda\) acts properly discontinuously on a contractible manifold \(K\setminus L\). The isotropy subgroups are finite and the orbit space \(K\setminus L/\Lambda\) is an orbifold. If \(\Lambda\) is torsion-free, then the action of \(\Lambda\) is free and the orbit space is a manifold. The purpose of this article is to prove a structure theorem for \(K\setminus L/\Lambda\); it roughly says that either it is a Riemannian orbifold of nonpositive sectional curvature or it Seifert fibers over such an orbifold. We do this if L satisfies the following extra condition: The center of \(MR\setminus L_ 0\) is finite, where \(L_ 0\) is the identity component of L, R is the radical of L, and M is the Lie subgroup of \(L_ 0\) which corresponds to the sum of the compact simple factors of the semi-simple semi-direct summand of a Levi decomposition of the Lie algebra of \(L_ 0\).