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

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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]).