On the automorphism group of parabolic structures and closed aspherical manifolds

TL;DR

This paper explores how parabolic geometric structures influence the topology and automorphism groups of closed aspherical manifolds, revealing strong restrictions and classifications for certain types like CR and quaternionic contact manifolds.

Contribution

It demonstrates that specific parabolic structures impose significant topological constraints and classifies automorphism groups for these manifolds, especially in the context of CR and quaternionic geometries.

Findings

Compact aspherical CR-manifolds with solvable fundamental groups are quotients of Heisenberg manifolds.

Parabolic structures restrict the topology of aspherical manifolds.

Automorphism groups are strongly influenced by the underlying parabolic geometry.

Abstract

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.