**Title:** Counting commensurability classes of hyperbolic manifolds

**Speaker:** Arie Levit

**Speaker Info:** Weizmann Institute

**Brief Description:**

**Special Note**:

**Abstract:**

An interesting direction in the study of hyperbolic manifolds is counting questions. By a classical result of Wang, in dimension > 3 there are finitely many isometry classes of 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 â€śmost" commensurability classes.

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.

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