Finite volume hyperbolic 3-manifolds whose fundamental group contains a subgroup that is locally free but not free

The purpose of this note is to show that the collection LF of finite volume hyperbolic 3-manifolds whose fundamental groups contain a subgroup that is locally free but not free is commensurably infinite. This result is stated formally as Theorem 4.1, and gives a strong answer to a question of Kropholler given in the Problem List in Niblo and Roller [19]. Recall that two hyperbolic 3-manifolds N1 and N2 are commensurable if there exists a hyperbolic 3manifold N that is a finite cover of both N1 and N2. Commensurability is an equivalence relation. We say that a collection M of hyperbolic 3-manifolds is commensurably infinite if it is infinite modulo the equivalence relation of commensurability. There are three main observations which make up the proof of Theorem 4.1. The first, discussed in Section 2, is that convex co-compact subgroups of a fundamental group of a hyperbolic 3-manifold N persist in the approximates given by hyperbolic Dehn surgery. This result is stated formally as Proposition 2.1. We then make use of the main result from [2], which gives a condition under which a collection of hyperbolic 3-manifolds is commensurably infinite. We state this result as Theorem 1.1.