Presentations for Subgroups of Artin Groups

Abstract

Recently, M. Bestvina and N. Brady have exhibited groups that are of type FP 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 rightangled 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 Fn of rank n, the group corresponding to the square is (F2), and the group corresponding to the octahedron is (F2). 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 FP (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 FP (3), which is H∆ in the case when ∆ is the octahedron [8], and R. Bieri’s group of type FP (n) but not of type FP (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, it is shown how to reduce this presentation to a finite one. This gives an independent and purely algebraic proof that H∆ is finitely presented when ∆ is simply connected. It would be interesting to give a similar proof of the converse. In Proposition 4 and Corollary 5, we review some Received by the editors May 17, 1997. 1991 Mathematics Subject Classification. Primary 20F36; Secondary 20E07, 20F32.

Cite this paper

@inproceedings{Dicks1998PresentationsFS, title={Presentations for Subgroups of Artin Groups}, author={Warren Dicks and IAN J. LEARY and Ronald Solomon}, year={1998} }