Toric complexes and Artin kernels

AbstractA simplicial complex L on n vertices determines a subcomplex TL of the n-torus, with fundamental group the right-angled Artin group GL. Given an epimorphism χ:GL→Z, let TLχ 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 TLχ with coefficients in a field k, viewed as modules over the group algebra kZ. We give combinatorial conditions for H⩽r(TLχ;k) to have trivial Z-action, allowing us to compute the truncated cohomology ring, H⩽r(TLχ;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