Toric complexes and Artin kernels

A simplicial complex L on n vertices determines a subcomplex T-L of the n-torus, with fundamental group the right-angled Artin group G(L). Given an epimorphism chi : G(L) --> Z, let T-L(chi) be the corresponding cover, with fundamental group the Artin kernel N-chi. We compute the cohomology jumping loci of the toric complex T-L, as well as the homology groups of T-chi with coefficients in a field k, viewed as modules over the group algebra kZ. We give combinatorial conditions for H (<= r)(T-L(chi); k) to have trivial Z-action, allowing us to compute the truncated cohomology ring, H-<= r(T-L(chi); k). We also determine several Lie algebras associated to Artin kernels, under certain triviality assumptions on the monodromy Z-action, andestablish the 1-formality of these (not necessarily finitely presentable) groups. (C) 2008 Elsevier Inc. All rights reserved.