THE BESTVINA–BRADY CONSTRUCTION REVISITED: GEOMETRIC COMPUTATION OF [sum ]-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[ratio ]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].