Bergeron–Venkatesh conjecture on torsion homology growth of arithmetic hyperbolic 3-manifolds

Suppose that

$$M_0 \xleftarrow{} M_1 \xleftarrow{} \dots$$

is a tower of covers of congruence arithmetic hyperbolic 3-manifolds whose injectivity radius approaches infinity. Write $H_1(M_i,\mathbb{Z})_{tors}$ for the torsion subgroup of the first homology group. Bergeron–Venkatesh conjecture. The torsion homology must grow exponentially in $\operatorname{vol}(M_i)$, and in fact

$$\lim_{i\to\infty}\frac{\log\lvert H_1(M_i,\mathbb{Z})_{tors}\rvert}{\operatorname{vol}(M_i)}=\frac{1}{6\pi}.$$

This conjecture predicts precise torsion homology growth for towers of congruence covers and connects topological torsion to spectral and number-theoretic phenomena. Its status is not resolved by the supplied context.