On the automorphism group of parabolic structures and closed aspherical manifolds

In this expository paper we discuss several properties on closed aspherical parabolic ${\sfG}$-manifolds $X/\Gamma$. These are manifolds $X/\Gamma$, where $X$ is a smooth contractible manifold with a parabolic ${\sfG}$-structure for which $\Gamma\leq \Aut_{\sfG}(X)$ is a discrete subgroup acting properly discontinuously on $X$ with compact quotient. By a parabolic $\sfG$-structure on $X$ we have in mind a Cartan structure which is modeled on one of the classical parabolic geometries arising from simple Lie groups $\sfG$ of rank one. Our results concern in particular the properties of the automorphism groups $\Aut_{\sfG}(X/\Gamma)$. Our main results show that the existence of certain parabolic ${\sfG}$-structures can pose strong restrictions on the topology of compact aspherical manifolds $X/\Gamma$ and their parabolic automorphism groups. In this realm we prove that any compact aspherical standard $CR$-manifold with virtually solvable fundamental group is diffeomorphic to a quotient of a Heisenberg manifold of complex type with its standard $CR$-structure. Furthermore we discuss the analogue properties of standard quaternionic contact manifolds in relation to the quaternionic Heisenberg group.