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 ), or there exists a nonaspherical subcomplex of an
aspherical 2-complex (a counterexample to the Whitehead Conjecture ). 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 ). Dicks and Leary  give an explicit way of obtaining presentations (on
finite generating sets) for these groups. For some of these examples, it is shown in
 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  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 ,
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.