The Bergeron–Venkatesh torsion growth conjecture for Bianchi groups

Let $\Gamma_0({\mathfrak p})$ be a congruence subgroup inside a fixed Bianchi group, where ${\mathfrak p}$ is a prime ideal of residue degree one. Write $N{\mathfrak p}$ for its norm, $H_1(-,\mathbb Z)_{tor}$ for the torsion subgroup of first homology, and $\operatorname{vol}$ for hyperbolic volume. Torsion growth conjecture.

$$\lim_{N{\mathfrak p} \rightarrow \infty} \dfrac{\log |H_1(\Gamma_0({\mathfrak p}),\mathbb Z)_{tor}|}{\operatorname{vol}(\Gamma_0({\mathfrak p}) \backslash \mathbb H)}=\dfrac{1}{6\pi},$$

where the limit is taken over prime ideals ${\mathfrak p}$ of residue degree one.

This extends the torsion-growth result of Bergeron and Venkatesh from cocompact arithmetic lattices and acyclic coefficient modules to the non-cocompact Bianchi setting with trivial coefficients. The statement is supported by numerical data, while the source points to more general conjectures for related towers.