The Bestvina-Brady Construction Revisited: Geometric Computation of ∑-Invariants for Right-Angled Artin Groups

Abstract

The starting point of our investigation is the remarkable paper [2] in which Bestvina and Brady gave an example of an infinitely related group of type FP2. The result about right-angled Artin groups behind their example is best interpreted by means of the Bieri–Strebel–Neumann–Renz Σ-invariants. For a group G the invariants Σn(G) and Σn(G, Z) are sets of non-trivial homomorphisms χ:G→R. They contain full information about finiteness properties of subgroups of G with abelian factor groups. The main result of [2] determines for the canonical homomorphism χ, taking each generator of the right-angled Artin group G to 1, the maximal n with χ ∈ Σn(G), respectively χ ∈ Σn(G, Z). In [6] Meier, Meinert and VanWyk completed the picture by computing the full Σ-invariants of right-angled Artin groups using as well the result of Bestvina and Brady as algebraic techniques from Σ-theory. Here we offer a new account of their result which is totally geometric. In fact, we return to the Bestvina–Brady construction and simplify their argument considerably by bringing a more general notion of links into play. At the end of the first section we re-prove their main result. By re-computing the full Σ-invariants, we show in the second section that the simplification even adds some power to the method. The criterion we give provides new insight on the geometric nature of the ‘n-domination’ condition employed in [6].