Alexander varieties and largeness of finitely presented groups

Abstract

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 \YN\ of Y which satisfy b1(YN)≥ N. We give some applications of this result to the study of hyperbolic 3--manifolds, mapping classes of surfaces and combinatorial group theory.

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