For more information about this meeting, contact Anatole Katok, Mari Royer, Yakov Pesin, Dmitri Burago.

Title: | Geometric consequences of algebraic rank in hyperbolic $3$-manifolds |

Seminar: | Center for Dynamics and Geometry Seminars |

Speaker: | Ian Biringer, Yale |

Abstract: |

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. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 03 / 31 / 2010 |

Time: | 03:30pm - 05:30pm |