Model aspherical manifolds with no periodic maps

Abstract

A. Borel proved that, if the fundamental group $E$ of an aspherical manifold $M$ is centerless and the outer automorphism group of $E$ is torsion–free, then $M$ admits no periodic maps, or equivalently, there are no non-trivial finite groups of homeomorphisms acting effectively on $M$. In the literature, taking off from this result, several examples of (rather complex) aspherical manifolds exhibiting this total lack of periodic maps have been presented. In this paper, we investigate to what extent the converse of Borel’s result holds for aspherical manifolds $M$ arising from Seifert fiber space constructions. In particular, for e.g. flat Riemannian manifolds, infra-nilmanifolds and infra-solvmanifolds of type (R), it turns out that having a centerless fundamental group with torsion–free outer automorphism group is also necessary to conclude that all finite groups of affine diffeomorphisms acting effectively on the manifold are trivial. Finally, we discuss the problem of finding (less complex) examples of such aspherical manifolds with no periodic maps.