Geometry and rank of fibered hyperbolic $3$–manifolds

Abstract

Recall that the rank of a finitely generated group is the minimal number of elements needed to generate it. In [Comm. Anal. Geom. 10 (2002) 377-395], M White proved that the injectivity radius of a closed hyperbolic $3$–manifold $M$ is bounded above by some function of $rank\left({\pi }_1\left(M\right)\right)$. Building on a technique that he introduced, we determine the ranks of the fundamental groups of a large class of hyperbolic $3$–manifolds fibering over the circle.