Incommensurability criteria for Kleinian groups

The purpose of this note is to present a criterion for an infinite collection of distinct hyperbolic 3-manifolds to be commensurably infinite. (Here, a collection of hyperbolic 3-manifolds is commensurably infinite if it contains representatives from infinitely many commensurability classes.) Namely, such a collection $\mathbf{M}$is commensurably infinite if there is a uniform upper bound on the volumes of the manifolds in $\mathbf{M}$.

There is a related criterion for an infinite collection of distinct finitely generated Kleinian groups with non-empty domain of discontinuity to be commensurably infinite. (Here, a collection of Kleinian groups is commensurably infinite if it is infinite modulo the combined equivalence relations of commensurability and conjugacy in $\operatorname{Isom}^+(\mathbf{H}^3)$.) Namely, such a collection $\mathbf{G}$ is commensurably infinite if there is a uniform bound on the areas of the quotient surfaces of the groups in $\mathbf{G}$.