The rank of the fundamental group of certain hyperbolic 3–manifolds fibering over the circle

Probably the most basic invariant of a finitely generated group is its rank, ie the minimal number of elements needed to generate it. In general the rank of a group is not computable. For instance, there are examples, due to Baumslag, Miller and Short [3], of hyperbolics groups showing that there is no uniform algorithm solving the rank problem. Everything changes in the setting of 3–manifold groups and recently Kapovich and Weidmann [9] gave an algorithm determining rank(π1(M)) when M is a 3–manifold with hyperbolic fundamental group. However, it is not possible to give a priori bounds on the complexity of this algorithm and hence it seems difficult to use it to obtain precise results in concrete situations. The goal of this note is to determine the rank of the fundamental group of a particularly nice class of 3–manifolds.