 Math. Gen.*]{} [**36**]{} 1213

[^1]: Such initial conditions imply that a SAW can traverse unit tetrahedrons along only one, or two nonconsecutive edges. This restriction of the standard SAW model does not alter the critical behavior of the system.


---
address:
- 'Rice University, Houston, Texas, 77005-1892'
- 'SUNY at Buffalo, Buffalo,N.Y.'
author:
- 'Tim D. Cochran'
- Joseph Masters
title: 'The Growth Rate of the First Betti Number in Abelian Covers of $3$-Manifolds '
---

[^1]

Abstract {#abstract .unnumbered}
========

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 \[IK\]).

Introduction {#introduction .unnumbered}
============

Motivated by Waldhausen’s work on Haken manifolds, and by W. Thurston’s [**Geometrization Conjecture**]{}, it has been variously conjectured that, if $M$ is an orientable, irreducible closed $3$-manifold with infinite fundamental group, then:

$M$ is finitely covered by a Haken manifold;

Some finite cover of $M$ has positive first Betti number;

Either $\pi_1(M)$ is virtually solvable or $M$ has finite covers with arbitrarily large first Betti number;

$M$ has a finite cover that fibers over the circle.

There are easy implications VIBNC$\Longrightarrow$VPBNC$\Longrightarrow$VHC and VFC$\Longrightarrow$VPBNC $\Longrightarrow$VHC. Each implies, if $M$ is atoroidal, the long-standing conjecture of Thurston that such a manifold admits a geometric structure. It is int