Alexander Stratifications of Character Varieties

There is a natural stratification of the character variety of a finitely presented group coming from the jumping loci of the first cohomology of one-dimensional representations. Equations defining the jumping loci 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 Simpson, Arapura and others show that if $\Gamma$ is the fundamental group of a compact K\"ahler manifold, then the strata are finite unions of translated affine subtori. If follows that for K\"ahler groups the jumping loci must be defined by binomial ideals. We discuss properties of the jumping loci of general finitely presented groups and apply the ``binomial criterion" to obtain new obstructions for one-relator groups to be K\"ahler.