In this joint work with Susan Williams we consider the conjecture: a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove it for a class of knots that includes all knots of genus 1. We also discuss two equivalent forms of the conjecture, one involving twisted Alexander polynomials, the other a weak form of subgroup separability. [Notes added by JAH. (1) This is in Carslaw 175, not the usual room. (2) Susan Williams shall be giving a JC on "Knots and Algebraic Dynamical Systems" (also joint work with Dan Silver) at UNSW, on Thursday 24 April. Details shall be posted separately.]