Finite group actions and cyclic branched covers of knots in $\mathbf{S}^3$

Abstract

We show that a hyperbolic $3$-manifold can be the cyclic branched cover of at most fifteen knots in $\mathbf{S}^3$. This is a consequence of a general result about finite groups of orientation preserving diffeomorphisms acting on $3$-manifolds. A similar, although weaker, result holds for arbitrary irreducible $3$-manifolds: an irreducible $3$-manifold can be the cyclic branched cover of odd prime order of at most six knots in $\mathbf{S}^3$.

31 pages, 1 figure. Changes from v2: The paper has been substantially reorganized, in particular the proof of Theorem 2 was considerably shortened. Accepted for publication by the Journal of Topology