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.

# Counting commensurability classes of hyperbolic manifolds (RTG Day)

Datum: | 22.11.2016 | Ort: | 1.067 (20.30) |
---|---|---|---|

Referent: | Arie Levit | ||

Zeit: | 13:30 Uhr |
||