# Meeting Details

Title: Geometric consequences of algebraic rank in hyperbolic $3$-manifolds Center for Dynamics and Geometry Seminars Ian Biringer, Yale Mostow's rigidity theorem states that a closed hyperbolic $3$-manifold M is determined up to isometry by the algebra of its fundamental group. We will discuss how the geometry of M is constrained by the minimal number of elements needed to generate its fundamental group; this invariant is called the (algebraic) rank of M. In particular, we will explain how M can be decomposed into a number of geometric building blocks such that the complexities of the blocks and of the decomposition depend only on M's algebraic rank and on a lower bound for M's injectivity radius. Our work links rank and injectivity radius to a number of other geometric invariants, including Heegaard genus, the Cheeger constant and the first eigenvalue of the Laplacian. One can also use the techniques involved to prove a finiteness statement for the number of commensurability classes of arithmetic closed hyperbolic $3$-manifolds with bounded rank and injectivity radius. Joint with Juan Souto.