Incommensurability criteria for Kleinian groups

The purpose of this note is to present a criterion for an infinite collecti on of distinct hyperbolic 3-manifolds to be commensurably infinite. (Here, a collection of hyperbolic 3-manifolds is commensurably infinite if it cont ains representatives from infinitely many commensurability classes.) Namely , such a collection M is commensurably infinite if there is a uniform upper bound on the volumes of the manifolds in M. There is a related criterion for an infinite collection of distinct finitel y generated Kleinian groups with non-empty domain of discontinuity to be co mmensurably infinite. (Here, a collection of Kleinian groups is commensurab ly infinite if it is infinite modulo the combined equivalence relations of commensurability and conjugacy in Isom + (H-3).) Namely, such a collection G is commensurably infinite if there is a uniform bound on the areas of the quotient surfaces of the groups in G.