The subexponential regulator conjecture for congruence arithmetic hyperbolic 3-manifolds

Let $M_k$ be a sequence of congruence arithmetic hyperbolic $3$-manifolds whose injectivity radii approach infinity. For each $M_k$, let $R(M_k)$ denote its regulator, defined from an $L^2$-basis of harmonic $1$-forms and a basis of $H_1(M_k,\mathbb{Z})$ by

$$R(M_k):=\det\left(\int_{\gamma_i}\omega_j\right)_{1\leq i,j\leq N}^{-2}.$$

Subexponential regulator conjecture.

$$\lim_{k\to\infty}\frac{\log R(M_k)}{\operatorname{vol}(M_k)}=0.$$

The regulator appears in the Cheeger–Müller formula alongside the torsion in $H_1(M_k,\mathbb{Z})$. Showing that it is subexponential in the volume would address one of the factors relevant to torsion growth, although controlling small Laplace eigenvalues remains a separate difficult issue.