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 $\mathbb{Z}_n$ covering spaces, $M_n$, in which $\beta_1(M_n)$ 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]).