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

TL;DR

This paper establishes a doubly exponential bound on the torsion in homology of closed hyperbolic manifolds and analyzes the growth of non-commensurable hyperbolic manifolds with respect to diameter, revealing that arithmetic manifolds become negligible at large diameters.

Contribution

It provides the first explicit bound on homological torsion in hyperbolic manifolds and quantifies the growth rate of non-commensurable manifolds, connecting to conjectures by Bergeron and Venkatesh.

Findings

Bound on torsion subgroups is doubly exponential in diameter

Number of non-commensurable manifolds grows rapidly with diameter

Fraction of arithmetic manifolds tends to zero as diameter increases

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.