Alexander varieties and largeness of finitely presented groups

TL;DR

This paper establishes a criterion linking the largeness of a finitely presented group's fundamental group to the existence of certain sequences of abelian covers with increasing first Betti numbers, with applications in topology and group theory.

Contribution

It provides a new characterization of largeness for finitely presented groups using finite covers and Betti number growth, connecting algebraic and topological properties.

Findings

Largeness of the fundamental group is equivalent to the existence of specific abelian cover sequences.

Applications include insights into hyperbolic 3-manifolds and surface mapping class groups.

The criterion offers a new tool for studying group largeness via topological covers.

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 \{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.