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The Bestvina–Brady Construction Revisited: Geometric Computation of ∑‐Invariants for Right‐Angled Artin Groups
Author(s) -
Bux KaiUwe,
Gonzalez Carlos
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799007991
Subject(s) - mathematics , abelian group , group (periodic table) , section (typography) , generator (circuit theory) , artin group , argument (complex analysis) , homomorphism , pure mathematics , von neumann architecture , algebraic number , computation , power (physics) , coxeter group , computer science , mathematical analysis , algorithm , chemistry , physics , biochemistry , organic chemistry , quantum mechanics , operating system
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 FP 2 . 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 Σ n ( G ) and Σ n ( 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 ∈ Σ n ( G ), respectively χ ∈ Σ n ( 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 ].