Homologically trivial actions on cyclic coverings of knots

Luisa Paoluzzi and Marco Reni

Vol. 188 (1999), No. 1, 155–177

DOI: 10.2140/pjm.1999.188.155

Abstract

A classical theorem of Hurewitz says that the isometry group of a closed 2-dimensional hyperbolic manifold acts faithfully on its first homology group. The analogous theorem in dimension 3 is false. In this paper we consider the class of 3-manifolds which are cyclic branched coverings of knots in the 3-sphere S³. We characterize the isometry group actions which are homologically faithful in the case of p-fold cyclic coverings of knots when p is sufficiently large. This characterization is given in terms of the knot polynomials.

Milestones

Received: 30 May 1997

Revised: 10 December 1997

Published: 1 March 1999