The Growth Rate of the First Betti Number in Abelian Covers of 3-manifolds

We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds M with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of M , for there to exist a family of Zn covering spaces, Mn, in which β1(Mn) increases linearly with n. The latter generalizes work of M. Katz and C. Lescop [KL], by showing that the non-vanishing of any one of these invariants of M is sufficient to guarantee certain optimal systolic inequalities for M (by work of Ivanov and Katz [IK]).