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 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 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 Isom+(H3).) 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. The purpose of this note is to explore conditions that imply that an infinite collection of distinct hyperbolic 3-manifolds is commensurably infinite. The main result, Theorem 0.1, is a criterion for a collection M of finite volume hyperbolic 3-manifolds to be commensurably infinite. We also present a related criterion, Theorem 0.4, for a collection G of finitely generated Kleinian groups with non-empty domain of discontinuity to be commensurably infinite. Recall that a Kleinian group is a discrete subgroup Γ of the group Isom(H3) of orientation preserving isometries of hyperbolic 3-space. A hyperbolic 3-manifold is the quotient of H by a (torsion-free) Kleinian group. It follows from the rigidity theorems of Mostow [9] and Prasad [10] that if N is a 3-manifold that admits a finite volume hyperbolic structure, then the realization of its fundamental group as a Kleinian group is unique up to conjugacy in Isom(H3). Two hyperbolic 3-manifolds N1 and N2 are commensurable if they have a common finite cover, that is, if there exists a hyperbolic 3-manifold N that is a finite cover of both N1 and N2. Commensurability is an equivalence relation on the set of hyperbolic 3-manifolds. A collection M of hyperbolic 3-manifolds is commensurably infinite if it contains representatives from infinitely many commensurability classes. Two Kleinian groups Γ1 and Γ2 are commensurable if their intersection Γ1 ∩ Γ2 has finite index in both Γ1 and Γ2. This definition is a bit weaker than that of hyperbolic 3-manifolds (in that commensurable Kleinian groups give rise to commensurable hyperbolic 3-manifolds, but not necessarily vice versa), and so we need 1991 Mathematics Subject Classification. Primary 57M50, 30F40; Secondary 20H10.