Counting commensurability classes of hyperbolic manifolds

TL;DR

This paper demonstrates that in dimensions four and higher, the number of non-arithmetic hyperbolic manifolds grows extremely rapidly with volume, indicating that most such manifolds are non-arithmetic.

Contribution

It establishes an exponential lower bound on the count of non-arithmetic hyperbolic manifolds up to commensurability in higher dimensions.

Findings

Number of hyperbolic manifolds of volume at most v is about v^v.

Almost all hyperbolic manifolds are non-arithmetic in high dimensions.

Method uses a geometric graph-of-spaces construction based on quadratic forms.

Abstract

Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v^v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms.