Toric Complexes and Artin Kernels

A simplicial complex L on n vertices determines a subcomplex TL of the ntorus, with fundamental group the right-angled Artin groupGL. Given an epimorphism χ : GL → Z, let T L be the corresponding cover, with fundamental group the Artin kernel Nχ. We compute the cohomology jumping loci of the toric complex TL, as well as the homology groups of T L with coefficients in a field k, viewed as modules over the group algebra kZ. We give combinatorial conditions for H≤r(T L ; k) to have trivial Z-action, allowing us to compute the truncated cohomology ring, H≤r(T L ; k). We also determine several Lie algebras associated to Artin kernels, under certain triviality assumptions on the monodromy Z-action, and establish the 1-formality of these (not necessarily finitely presentable) groups.