Alexander stratifications of character varieties

Abstract

Equations defining the jumping loci for the first cohomology group of one-dimensional representations of a finitely presented group Γ can be effectively computed using Fox calculus. In this paper, we give an exposition of Fox calculus in the language of group cohomology and in the language of finite abelian coverings of CW complexes. Work of Arapura and Simpson imply that if Γ is the fundamental group of a compact Kähler manifold, then the strata are finite unions of translated affine subtori. It follows that for Kähler groups the jumping loci must be defined by binomial ideals. As we will show, this is not the case for general finitely presented groups. Thus, the “binomial condition” can be used as a criterion for proving certain finitely presented groups are not Kähler.