Geometric realization of a finite subgroup of 𝜋₀𝜖(𝑀). II

Abstract

Let M M be a closed aspherical manifold with a virtually 2 2 -step nilpotent fundamental group. Then any finite group G G of homotopy classes of self-homotopy equivalences of M M can be realized as an effective group of self-homeomorphisms of M M if and only if there exists a group extension E E of π \pi by G G realizing G → Out π 1 M G \to {\operatorname {Out }}{\pi _1}M so that C E ( π ) {C_E}(\pi ) , the centralizer of π \pi in E E , is torsion-free. If this is the case, the action ( G , M ) (G,M) is equivalent to an affine action ( G , M ′ ) (G,M’) on a complete affinely flat manifold homeomorphic to M M . This generalizes the same result for flat Riemannian manifolds.