  Every compact aspherical Riemannian manifold admits a canonical series of
orbibundle structures with infrasolv fibers which is called its infrasolv
tower. The tower arises from the solvable radicals of isometry group actions on
the universal covers. Its length and the geometry of its base measure the
degree of continuous symmetry of an aspherical Riemannian manifold. We say that
the manifold has large symmetry if it admits an infrasolv tower whose base is a
locally homogeneous space. We construct examples of aspherical manifolds with
large symmetry, which do not support any locally homogeneous Riemannian
metrics.
