Abstract
Let M be a closed aspherical manifold and A a finite subgroup of the outer automorphism group Out π1M of π1M. A necessary (and in many cases also sufficient) condition for realising A by the induced action of an isomorphic group of homeomorphisms of M is the existence of an extension 1→π1M→E→A→1 to the abstract kernel (A,π1M, A↪Out π1M). If the center of π1M is nontrivial, this condition need not be fulfilled ([14]). We showed in [25] however that one can always find a surjection B→A of a finite group B with abelian kernel such that there exists an extension to the abstract kernel (B,π1M,B→A↪Outπ1M), and one can try to realize B instead of A. The main result of the present paper is a characterisation of all such groups B (for a given A) which can be realized by a group of homeomorphisms. The class of manifolds considered here consists of certain Seifert fiber spaces in arbitrary dimensions but the main result is purely algebraic and can be applied to other classes of manifolds, for example to flat Riemannian manifolds.
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Zimmermann, B. Nielsensche Realisierungssätze für Seifertfaserungen. Manuscripta Math 51, 225–242 (1985). https://doi.org/10.1007/BF01168354
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DOI: https://doi.org/10.1007/BF01168354