Geometric realization of a finite subgroup of $\pi _{0}\varepsilon (M)$. II

Abstract:

Let $M$ be a closed aspherical manifold with a virtually $2$-step nilpotent fundamental group. Then any finite group $G$ of homotopy classes of self-homotopy equivalences of $M$ can be realized as an effective group of self-homeomorphisms of $M$ if and only if there exists a group extension $E$ of $\pi$ by $G$ realizing $G \to {\operatorname {Out }}{\pi _1}M$ so that ${C_E}(\pi )$, the centralizer of $\pi$ in $E$, is torsion-free. If this is the case, the action $(G,M)$ is equivalent to an affine action $(G,M’)$ on a complete affinely flat manifold homeomorphic to $M$. This generalizes the same result for flat Riemannian manifolds.