Bestvina–Brady groups and the plus construction

Abstract

A recent result of Bestvina and Brady [1, theorem 8·7], shows that one of two outstanding questions has a negative answer; either there exists a group of cohomological dimension 2 and geometric dimension 3 (a counterexample to the Eilenberg–Ganea Conjecture [4]), or there exists a nonaspherical subcomplex of an aspherical 2-complex (a counterexample to the Whitehead Conjecture [11]). More precisely, Bestvina and Brady construct a family of groups which are potential counterexamples to the Eilenberg–Ganea Conjecture, each of which has cohomological dimension 2. These are also examples of groups of type FP₂ which are not finitely presented (see [1]). Dicks and Leary [3] give an explicit way of obtaining presentations (on finite generating sets) for these groups. For some of these examples, it is shown in [1] that any 2-dimensional classifying space would give rise to a counterexample to the Whitehead conjecture.

We will refer to the examples cited above as Bestvina–Brady groups. These come equipped with natural, nonpositively curved cubical 3-dimensional classifying complexes, which we will call Bestvina–Brady complexes. In this short note, we show that these Bestvina–Brady complexes are (up to homotopy equivalence) formed by applying the Quillen plus construction to certain finite 2-complexes. From this, together with known facts about 2-complexes with aspherical plus constructions, we recover the result of Bestvina and Brady [1] that the Bestvina–Brady groups act freely on acyclic 2-complexes, and hence have cohomological dimension at most 2. It also follows that these groups have free relation modules of finite rank and so are of type FF. Finally, we use our construction to give an alternative proof of the cited theorem of Bestvina and Brady; at least one of the Eilenberg–Ganea and Whitehead conjectures is false.

We do not use the full force of the Morse-theoretical techniques developed in [1], but will assume two results form that paper; the asphericity of the Bestvina–Brady complexes and the non-finite presentability of the Bestvina–Brady groups.