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

### Volume 184 / 2004

#### 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).