We are interested in finite groups acting orientation-preservingly on
3–manifolds (arbitrary actions, ie not necessarily free
actions). In particular we consider finite groups which contain an
involution with nonempty connected fixed point set. This condition is
satisfied by the isometry group of any hyperbolic cyclic branched
covering of a strongly invertible knot as well as by the isometry
group of any hyperbolic 2–fold branched covering of a knot in
S3. In the paper we give a characterization of nonsolvable
groups of this type. Then we consider some possible applications to
the study of cyclic branched coverings of knots and of hyperelliptic
diffeomorphisms of 3–manifolds. In particular we analyze the
basic case of two distinct knots with the same cyclic branched
covering.
To the memory of Heiner
Zieschang
Keywords
3-manifold, finite group action, cyclic
branched covering, knot in S³
