Algebraic Criteria to Decide if a Finite Group Acts Effectively on a Model Aspherical Manifold

For an aspherical manifoldM, arising from a Seifert fiber space construction, it is known that, under some additional conditions onM, a finite abstract kernel ψ:F → Out(π1(M)) can be (effectively) geometrically realized by a group of fiber preserving homeomorphisms ofMif and only if ψ can be realized by an (admissible) group extension 1 → π1(M) → E → F → 1. Hence, the study of the symmetry of such a manifold (in terms of finite effective actions onM) can be converted into a group-theoretical study of realizing (algebraically) finite abstract kernelsF → Out(π1(M)). This question, conceptually, is well understood: there exists an extension realizing a given abstract kernel if and only if the corresponding third cohomology class, called the obstruction, vanishes. Unfortunately, a straightforward computation of this obstruction can be extremely hard. In this paper, we present some criteria which solve this problem for certain finite abstract kernels. Instead of using cohomological arguments, a completely different, rather technical and computational approach, based on the Reidemeister–Schreier method for presenting subgroups of finite index in a given finitely presented group, is followed. Not only the actual results but also this approach is of interest since it certainly allows one to produce similar criteria for other finite groups