Alexander varieties and largeness of finitely presented groups

Let X be a finite CW complex. We show that the fundamental group of X is large if and only if there is a finite cover Y of X and a sequence of finite abelian covers {Y_N} of Y which satisfy b_1(Y_N)\geq N. We give some applications of this result to the study of hyperbolic 3–manifolds, mapping classes of surfaces and combinatorial group theory.