Introduction

By a hyperbolic 3-manifold we shall always mean a complete orientable hyperbolic 3-manifold of finite volume. A hyperbolic 3-manifold M is said to be n-generator if the minimal number of elements required to generate π1(M) is n. The focus of this paper is 2-generator hyperbolic 3-manifolds, the main aim being to give a construction of infinitely many closed hyperbolic 3-manifolds which are not 2-generator, but have a proper finite cover which is. Our interest in such examples was motivated by the deep results contained in [4] and [10] which relate questions on 2-generator subgroups of hyperbolic 3-manifold groups to estimates on the lower bound for the smallest volume of a closed hyperbolic 3-manifold. We also construct certain 2-generator Haken hyperbolic 3-manifolds whose existence helps to explain why more recent methods of Culler and Shalen (in preparation) seem to be necessary for estimating volumes of closed Haken hyperbolic 3-manifolds.

To describe the connection between the articles referred to above and the contents of this article we need to recall the definition of a Margulis number of a hyperbolic 3-manifold.

Let M = H/Γ be a closed hyperbolic 3-manifold and ò > 0. Then ò is a Margulis number for M if for every point z of H and every pair of noncommuting elements γ and δ of Γ we have max{ξ(z,γ(z)),ξ(z,δ(z))} ≥ ò, where ξ denotes the hyperbolic metric.