Counting commensurability classes of hyperbolic manifolds

Tsachik Gelander; Arie Levit

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arXiv · 1401.8003

Counting commensurability classes of hyperbolic manifolds

Tsachik Gelander·Arie Levit

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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.

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BibTeX RIS

Tsachik Gelander, Arie Levit. 2014-05-20. Counting commensurability classes of hyperbolic manifolds. https://arxiv.org/abs/1401.8003

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