Cyclic branched coverings and homology 3-spheres with large group actions

Volume 184 / 2004

Bruno P. Zimmermann Fundamenta Mathematicae 184 (2004), 343-353 MSC: 57M12, 57M25, 57M50, 57M60, 57S17. DOI: 10.4064/fm184-0-19

Abstract

We show that, if the covering involution of a $3$-manifold $M$ occurring as the $2$-fold branched covering of a knot in the $3$-sphere is contained in a finite nonabelian simple group $G$ of diffeomorphisms of $M$, then $M$ is a homology $3$-sphere and $G$ isomorphic to the alternating or dodecahedral group ${\mathbb A}_5 \cong {\rm PSL}(2,5)$. An example of such a $3$-manifold is the spherical Poincaré sphere. We construct hyperbolic analogues of the Poincaré sphere. We also give examples of hyperbolic $\mathbb Z_2$-homology $3$-spheres with ${\rm PSL}(2,q)$-actions, for various small prime powers ,$q$. We note that the groups ${\rm PSL}(2,q)$, for odd prime powers ,$q$, are the only candidates for being finite nonabelian simple groups which possibly admit actions on $\mathbb Z_2$-homology $3$-spheres (but the exact classification remains open).

Authors

  • Bruno P. ZimmermannDipartimento di Matematica e Informatica
    Università degli Studi di Trieste
    34100 Trieste, Italy
    e-mail

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