Geometry and Rank of Fibered
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چکیده
Recall that the rank of a finitely generated group is the minimal number of elements needed to generate it. M. White [Whi02] has proven that the injectivity radius of M is bounded above by some function of rank(π1(M)). Building on a technique that he introduced, we show that if M is a hyperbolic 3-manifold fibering over the circle with fiber Σg and rank(π1(M)) 6= 2g + 1, then the diameter of M is bounded above by some function of its injectivity radius. Equivalently, after fixing g and there are at most finitely many such M for which rank(π1(M)) 6= 2g + 1. Let Σg be the closed orientable surface of genus g and φ : Σg → Σg a homeomorphism. We can construct a 3-manifoldMφ, called the mapping torus of φ, as the quotient space
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