Abstract
Associated to every compact 3-manifold M and positive integer b, there is a constant c( M, b) = c. Any collection F i of incompressible surfaces with Betti numbers b 1 F i < 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.
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