Counting commensurability classes of hyperbolic manifolds (RTG Day)

  • Date:22.11.2016
  • Speaker:Arie Levit
  • Time:13:30 h
  • Abstract: Counting questions are an interesting direction in the study of hyperbolic manifolds. By a classical result of Wang, in dimension > 3 there are only finitely many hyperbolic manifolds up to any finite volume V. More recently, Burger, Gelander, Lubotzky and Mozes showed that this number grows like V^V.
    In this talk we focus on the number of commensurability classes of hyperbolic manifolds. Two manifolds are commensurable if they admit a common finite cover. We show that in dimension > 3 this number grows like V^V as well.
    Since the number of arithmetic commensurability classes grows ~ polynomially, our result implies that non-arithmetic manifolds account for “most" commensurability classes. Another application has to do with quasi-isometry classes of lattices.
    We will explain the ideas involved in the proof, which include a mixture of arithmetic, hyperbolic geometry and some combinatorics.
    This is a joint work with Tsachik Gelander.

  • Place:1.067 (20.30)