This post and the next will be mostly based on a 1996 paper by Mladen Bestvina and Noel Brady, Morse theory and finiteness properties of groups. Their main result (the Bestvina–Brady Theorem) relates topological properties of a flag complex to the finiteness properties of a certain subgroup of a corresponding right-angled Artin group. In the current post, we will discuss applications of the Bestvina–Brady Theorem, while in the next, we will sketch the proof of this theorem. A reference for these applications as well as other open problems is the talk slides Outstanding Problems in Low-Dimensional Topology and Group Theory by Tim Riley.
Let be a finite flag complex — that is, a finite simplicial complex completely determined by its -skeleton in that it contains a simplex for each collection of pairwise adjacent vertices. Let be the collection of vertices of and the collection of edges of . The associated right-angled Artin group is
Consider the natural epimorphism , sending each generator to . The kernel of is called the Bestvina–Brady group associated to . The Bestvina-Brady Theorem relates finiteness properties of to topological properties of .
Before stating the Bestvina–Brady Theorem, we define some terms. We denote an Eilenberg–MacLane space for a group by .
Consider a group .
- is of type if there exists a space with finite -skeleton.
- is of type if admits a partial projective resolution
by finitely generated projective -modules.
- is of type if admits a projective resolution
by finitely generated projective -modules.
Observe that a group is of type if and only if it is finitely generated, and is of type if and only if it is finitely presented. Let us now consider the relationships among the three notions above. Clearly, implies for all .
Claim. implies for all .
Proof: Let be of type and let be the associated complex with finite -skeleton. Let be the universal cover of and consider its cellular chain complex,
Observe that each is a free -module with basis in one–to–one correspondence with -cells of . Also, as is the universal cover of a space, it is contractible and hence for all . Hence, the cellular chain complex forms an exact sequence. Consequently, we obtain a free resolution of over , where the first terms in the sequence are finitely generated.
It is proved in Ken Brown’s Cohomology of groups [Chap. 8, Sect. 7] that for , a finitely presented group is of type must also be of type . We are now ready to state the theorem. Recall that a topological space is said to be homologically -connected if the reduced homology groups, , are trivial for all , and a space is said to be acyclic if all of its reduced homology groups are trivial.
The Bestvina–Brady Theorem. Let be a finite flag complex, the associated right-angled Artin group, and the corresponding Bestvina–Brady group. Then
- if and only if is homologically -connected.
- if and only if is acyclic.
- is finitely presented if and only if is simply-connected.
Again, we will sketch a proof in the next post. Let us now consider some applications of this theorem.
Application 1: Bieri–Stallings groups
As a consequence of the Bestvina–Brady Theorem, for each we can provide a simple example of a group that is of type but not of type . To see this, consider , triangulated as an -fold join of -spheres. We get — that is, is the direct product of free groups on generators. Note that and for all . Thus, from the Bestvina–Brady Theorem we can conclude that is of type since is homologically -connected, but is not of type since is not homologically -connected. In this example, the are exactly the Bieri–Stallings groups discussed previously.
Application 2: the Relation Gap Problem
For this and the next application, we will consider an acyclic non-simply-connected finite flag complex of dimension . One example of such a flag complex is a flag triangulation of a spine of the Poincaré homology sphere — the unique -manifold with homology groups of a -sphere and finite non-trivial fundamental group. (HenrikRuep‘s video journey through the Poincaré homology sphere, complete with musical accompaniment, is available below.) The Bestvina–Brady Theorem implies that is of type (hence is of type ), but is not finitely presented.
Consider a presentation
where is free on the generators and — that is, is the normal closure of the subgroup generated by the elements . Note that acts on by conjugation, and hence induces an action of on . On account of this action,
is an upper bound on the rank of as a -module. The difference between this upper bound and the actual rank is called the relation gap. It is an open problem, whether there exists a group with a finite non-zero relation gap.
However, the Bestvina–Brady Theorem combines with proposition below to yield a group with an infinitely large relation gap. Let , where is a free group of finite rank . Then is of type iff is finitely generated as a -module. A proof of this proposition based on Schanuel’s Lemma from homological algebra can be found in Michael Tweedale’s Thesis.
Now, let be an acyclic non-simply-connected finite flag complex of dimension . As noted in the remark above, is of type but not of type . Consequently, the rank of as a -module is finite but is not finitely presented, leading to an infinite relation gap for .
Application 3: Eilenberg–Ganea versus Whitehead
The Bestvina–Brady group arising from a spine of a Poincaré Homology Sphere (described above) gives a counterexample to either the Eilenberg–Ganea Conjecture or to the Whitehead Conjecture. It is not known which. To state these conjectures, we first recall some definitions.
The geometric dimension, , of is the minimal such that there is a for of dimension .
The cohomological dimension, , of is the minimal such that admits a resolution by projective -modules.
Note that for any group , we have since the cellular chain complex of yields the desired projective resolution of .
Theorem (Stallings–Swan). is free.
Theorem (Eilenberg–Ganea). If , then .
So whenever .
Conjecture (Eilenberg–Ganea). implies .
Recall that a topological space is said to be aspherical if all of its higher homotopy groups are trivial.
Conjecture (Whitehead). Every connected subcomplex of an aspherical 2-complex is aspherical.
Let be the spine of the Poincaré Homology sphere as above. It is known that . Bestvina and Brady showed that if , as per the Eilenberg–Ganea Conjecture, then there exists a space contradicting the Whitehead Conjecture. Thus, one of these conjectures must be false.