Geometric consequences of algebraic rank in hyperbolic 3-manifolds

Ian Biringer, Yale

Mostow's rigidity theorem implies that the geometry of a closed hyperbolic 3-manifold M is completely determined by the isomorphism type of its fundamental group G. We will describe how rank(G), the minimal size of a generating set for G, influences the geometry of M. Under an assumption on injectivity radius, we will see that M can be constructed by gluing together copies of a bounded number of geometric pieces in some configuration whose complexity is controlled by rank(G). Applications to Heegaard genus, virtual fibering and arithmetic manifolds will be given. Joint with Juan Souto.