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 𝔸₅ ≅ 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 ℤ₂-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 ℤ₂-homology 3-spheres (but the exact classification remains open).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm184-0-19,
author = {Bruno P. Zimmermann},
title = {Cyclic branched coverings and homology 3-spheres with large group actions},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {343-353},
zbl = {1070.57002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm184-0-19}
}
Bruno P. Zimmermann. Cyclic branched coverings and homology 3-spheres with large group actions. Fundamenta Mathematicae, Tome 184 (2004) pp. 343-353. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm184-0-19/