Presentations for Subgroups of Artin Groups


Recently, M. Bestvina and N. Brady have exhibited groups that are of type F P but not finitely presented. We give explicit presentations for groups of the type considered by Bestvina-Brady. This leads to algebraic proofs of some of their results. Let ∆ be a finite flag complex, that is, a finite simplicial complex that contains a simplex bounding every complete subgraph of its 1-skeleton. The associated right-angled Artin group G ∆ is the group given by the presentation with generators the vertex set of ∆, and relators the commutators [v, w] for each pair of adjacent vertices in ∆. For example, the n-simplex corresponds to a free abelian group of rank n+1, a complex consisting of n points corresponds to the free group F n of rank n, the group corresponding to the square is (F 2) 2 , and the group corresponding to the octahedron is (F 2) 3. Provided that ∆ is non-empty, there is a homomorphism from G ∆ onto the integers, that takes every generator to 1. The group H ∆ is defined to be the kernel of this homomorphism. Remarkable recent work of Mladen Bestvina and Noel Brady has shown that the homological finiteness properties of the group H ∆ are controlled by the topology of the complex ∆ [1]. They show that H ∆ is finitely generated if and only if ∆ is connected, H ∆ is finitely presented if and only if ∆ is 1-connected, and H ∆ is of type F P (n) if and only if ∆ is (n− 1)-acyclic. Precursors of this result include J. Stallings' group that is finitely presented but not of type F P (3), which is H ∆ in the case when ∆ is the octahedron [8], and R. Bieri's group of type F P (n) but not of type F P (n + 1), which is H ∆ in the case when ∆ is a join of (n + 1) pairs of points (section 2.6 of [2]). The arguments used by Bestvina and Brady are geometric, and they do not give presentations for the groups that they consider. Theorem 1 of this paper gives a presentation for H ∆ for any connected ∆. The generators in the presentation are the edges of ∆, and each 1-cycle in ∆ gives rise to an infinite family of relators. In the case when ∆ is simply connected, …

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