Bestvina-Brady Groups and the Plus Construction

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, they consruct 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 nitely presented (see [1]). For one of these examples, they show 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 nite 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 nite 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.