Counting non-commensurable hyperbolic manifolds and a bound on homological torsion

Bram Petri

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arXiv · 1709.01873

Counting non-commensurable hyperbolic manifolds and a bound on homological torsion

Bram Petri

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Abstract

We prove that the cardinality of the torsion subgroups in homology of a closed hyperbolic manifold of any dimension can be bounded by a doubly exponential function of its diameter. It would follow from a conjecture by Bergeron and Venkatesh that the order of growth in our bound is sharp. We also determine how the number of non-commensurable closed hyperbolic manifolds of dimension at least 3 and bounded diameter grows. The lower bound implies that the fraction of arithmetic manifolds tends to zero as the diameter goes up.

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Bram Petri. 2017-09-06. Counting non-commensurable hyperbolic manifolds and a bound on homological torsion. https://arxiv.org/abs/1709.01873

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