Maximal toral action on aspherical manifolds \(\Gamma\) \(\setminus G/K\) and G/H (Q1114974)

In this note, we shall consider only topological actions. For a closed aspherical manifold M, it is well known that if a compact connected Lie group G acts on M effectively, then G is a toral group \(T^ s\) with \(s\leq rank\) of the center \(z(\pi_ 1(M))\) of the fundamental group \(\pi_ 1(M)\) of M [\textit{P. E. Conner} and \textit{F. Raymond}, Topology of Manifolds, Proc. Univ. Georgia 1969, 227-264 (1971; Zbl 0312.57025)]. In Bull. Am. Math. Soc. 83, 36-85 (1977; Zbl 0341.57003), \textit{P. E. Conner} and \textit{F. Raymond} conjectured that if M is a closed aspherical manifold, then (1) \(z(\pi_ 1(M))\) is finitely generated, say of rank k, (2) there exists a toral group \(T^ k\) acting effectively on M. These have been verified in many cases. For example, if M is a smooth manifold admitting a Riemannian metric with non-positive sectional curvature or if M is a nilmanifold, then (1) and (2) hold [see \textit{K. B. Lee} and \textit{F. Raymond}, Contemp. Math. 36, 367-425 (1985; Zbl 0564.57002)]. In this note, we shall prove the following: Theorem A. The conjectures (1) and (2) hold for aspherical manifolds of type \(\Gamma\) \(\setminus G/K\), where G is a connected non-compact Lie group, K a maximal compact subgroup of G and \(\Gamma\) a torsion free discrete uniform subgroup of G. Theorem B. The conjectures (1) and (2) hold for a compact homogeneous aspherical manifold G/H, where G is a connected non-compact Lie group and H a closed subgroup of G.