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

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 A(5) congruent to PSL(2,5). An example of such a 3-manifold is the spherical Poincare sphere. We construct hyperbolic analogues of the Poincare sphere. We also give examples of hyperbolic Z(2)-homology 3-spheres with PSL(2, q)-actions, for various small prime powers q. We note that the groups PSL(2, q), for odd prime powers q, are the only candidates for being finite nonabelian simple groups which possibly admit actions on Z(2)-homology 3-spheres (but the exact classification remains open).