Finite groups acting on 3-manifolds and cyclic branched coverings of knots

Details Finite groups acting on 3-manifolds and cyclic branched coverings of knots Finite groups acting on 3-manifolds and cyclic branched coverings of knots

2009-04-12

arxiv.org/abs/0904.1854v1

Geom. Topol. Monogr. 14 (2008) 393-416

Mathematics

Geometric Topology

This is the version published by Geometry & Topology Monographs on 29 April 2008

Scientific paper

10.2140/gtm.2008.14.393

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 the 3-sphere. 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.

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