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

TL;DR

This paper characterizes nonsolvable finite groups acting on 3-manifolds with involutions having nonempty fixed points, and explores applications to cyclic branched coverings of knots and hyperelliptic diffeomorphisms.

Contribution

It provides a characterization of nonsolvable groups acting on 3-manifolds with specific involutions, advancing understanding of symmetries in knot theory and 3-manifold topology.

Findings

Characterization of nonsolvable groups with involutions on 3-manifolds.

Analysis of cyclic branched coverings of knots.

Insights into hyperelliptic diffeomorphisms of 3-manifolds.

Abstract

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.