Toric complexes and Artin kernels

Abstract

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\colon G_L\to \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_L^\chi with coefficients in a field \k, viewed as modules over the group algebra \k\Z. We give combinatorial conditions for H_{\le r}(T_L^\chi;\k) to have trivial \Z-action, allowing us to compute the truncated cohomology ring, H^{\le r}(T_L^\chi;\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.Comment: 34 page

Bieri-Neumann-Strebel-Renz invariants and homology jumping loci

“…Jumping loci of toric complexes. The resonance and characteristic varieties of toric complexes were computed in [46], extending previous results from [44] and [22]. The cohomology group H 1 (T L , k) = k L 1 may be identified with the k-vector space with basis V, denoted k V .…”

“…In previous work [46], we showed that the triviality of the monodromy action of Z on i≤k H i (N χ , Q) can be tested in a purely combinatorial way. Moreover, if H 1 (N χ , Q) is a trivial QZ-module, then N χ is finitely generated.…”

“…The Σ-invariants of such groups (with coefficients in an arbitrary commutative ring), were computed by Meier, Meinert, and VanWyk in [39], with a more convenient description given later on by Bux and Gonzalez in [13]. The characteristic and resonance varieties of a toric complex T L were computed in [46], in terms of the algebraic combinatorics of the underlying simplicial complex L. In particular, the exponential formula holds for a toric complex, in arbitrary degree and depth. By Theorem 1.1(2),…”

“…The Aomoto-Betti numbers of the Stanley-Reisner ring may be computed solely in terms of the simplicial complex L, by means of the following formula from Aramova, Avramov, and Herzog [1], as slightly modified in [46]:…”

The spectral sequence of an equivariant chain complex and homology with local coefficients

“…In [24], we analyze the monodromy action on the homology of Galois Z-covers, for toric complexes associated to finite simplicial complexes. Using Part (2) of the Theorem, we obtain a combinatorial criterion for the full triviality of this action, up to a given degree.…”