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.