Hyperbolic 3-manifolds with k-free fundamental group

We prove that if M   is a closed, orientable, hyperbolic 3-manifold such that all subgroups of π1(M)π1(M) of rank at most k=5k=5 are free, then one can choose a point P in M   such that the set of all elements of π1(M,P)π1(M,P) that are represented by loops of length less than log9 is contained in a subgroup of rank at most 2; in particular, they generate a free group. In the 1990s, Culler, Shalen, and their co-authors initiated a program to understand the relationship between the topology and geometry of a closed hyperbolic 3-manifold; this paper extends those results to the setting of hyperbolic 3-manifolds with k=5k=5-free fundamental group. A key ingredient in the proof is an analogue of a group-theoretic result of Kent and Louder–McReynolds concerning intersections and joins of rank three subgroups of a free group. Moreover, we state conjectural extensions of the 5-free result for values k>5k>5, and establish them modulo the conjectured extension of the Kent and Louder–McReynolds result.