Kneser-Haken finiteness for bounded 3-manifolds, locally free groups, and cyclic covers

Associated to every compact 3-manifold M and positive integer b, there is a constant c(M,b) = c. Any collection Fi of incompressible surfaces with Betti numbers b1Fi < b for all i, none of which is a boundary parallel annulus or a boundary parallel disk, and no two of which are parallel, must have fewer than c members. Our estimate for c is exponential in b. This theorem is used to detect closed incompressible surfaces in the infinite cyclic covers of all non-fibered knot complements. In other terms, if the commutator subgroup of a knot group is locally free, then it is actually a finitely generated free group.