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 ${L}$ be a finite flag complex — that is, a finite simplicial complex completely determined by its ${1}$-skeleton in that it contains a simplex for each collection of pairwise adjacent vertices. Let ${L^{(0)}=\{v_1,\ldots, v_N\}}$ be the collection of vertices of ${L}$ and ${L^{(1)}}$ the collection of edges of ${L}$. The associated right-angled Artin group is

$\displaystyle G_L=\langle g_1, \ldots,g_N: [g_i,g_j]=1,\forall (v_i, v_j)\in L^{(1)}\rangle.$

Consider the natural epimorphism ${\phi\colon G_L\rightarrow \mathbb{Z}}$, sending each generator ${g_i}$ to ${1\in\mathbb{Z}}$. The kernel ${H_L}$ of ${\phi}$ is called the Bestvina–Brady group associated to ${L}$. The Bestvina-Brady Theorem relates finiteness properties of ${H_L}$ to topological properties of ${L}$.

Before stating the Bestvina–Brady Theorem, we define some terms. We denote an Eilenberg–MacLane space for a group ${\Gamma}$ by ${K(\Gamma, 1)}$.

Consider a group ${\Gamma}$.

1. ${\Gamma}$ is of type ${F_n}$ if there exists a ${K(\Gamma, 1)}$ space with finite ${n}$-skeleton.
2. ${\Gamma}$ is of type ${\textit{FP}_n}$ if ${\mathbb{Z}}$ admits a partial projective resolution

$\displaystyle P_n\rightarrow \ldots\rightarrow P_1\rightarrow P_0\rightarrow \mathbb{Z}\rightarrow 0$

by finitely generated projective ${\mathbb{Z}\Gamma}$-modules.

3. ${\Gamma}$ is of type ${\textit{FP}}$ if ${\mathbb{Z}}$ admits a projective resolution

$\displaystyle 0\rightarrow P_n\rightarrow \ldots\rightarrow P_1\rightarrow P_0\rightarrow \mathbb{Z}\rightarrow 0$

by finitely generated projective ${\mathbb{Z}\Gamma}$-modules.

Observe that a group is of type ${F_1}$ if and only if it is finitely generated, and is of type ${F_2}$ if and only if it is finitely presented. Let us now consider the relationships among the three notions above. Clearly, ${\textit{FP}}$ implies ${\textit{FP}_n}$ for all ${n}$.

Claim. ${F_n}$ implies ${\textit{FP}_n}$ for all ${n}$.

Proof: Let ${\Gamma}$ be of type ${F_n}$ and let ${X}$ be the associated ${K(\Gamma, 1)}$ complex with finite ${n}$-skeleton. Let ${\tilde{X}}$ be the universal cover of ${X}$ and consider its cellular chain complex,

$\displaystyle \cdots \rightarrow C_n(\tilde{X})\rightarrow C_{n-1}(\tilde{X})\rightarrow \cdots \rightarrow C_1(\tilde{X})\rightarrow C_0(\tilde{X})\rightarrow \mathbb{Z}\rightarrow 0.$

Observe that each ${C_i(\tilde{X})}$ is a free ${\mathbb{Z}\Gamma}$-module with basis in one–to–one correspondence with ${i}$-cells of ${X}$. Also, as ${\tilde{X}}$ is the universal cover of a ${K(\Gamma, 1)}$ space, it is contractible and hence ${\widetilde{H}_i(\tilde{X})=0}$ for all ${i\geq 0}$. Hence, the cellular chain complex forms an exact sequence. Consequently, we obtain a free resolution of ${\mathbb{Z}}$ over ${\mathbb{Z}\Gamma}$, where the first ${n}$ terms in the sequence are finitely generated. $\Box$

It is proved in Ken Brown’s Cohomology of groups [Chap. 8, Sect. 7] that for ${n\geq 2}$, a finitely presented group is of type ${\textit{FP}_n}$ must also be of type ${F_n}$. We are now ready to state the theorem. Recall that a topological space ${L}$ is said to be homologically ${n}$-connected if the reduced homology groups, ${\widetilde{H}_i(L)}$, are trivial for all ${i\leq n}$, and a space ${L}$ is said to be acyclic if all of its reduced homology groups are trivial.

The Bestvina–Brady Theorem. Let ${L}$ be a finite flag complex, ${G_L}$ the associated right-angled Artin group, and ${H_L\leq G_L}$ the corresponding Bestvina–Brady group. Then

1. ${H_L\in \textit{FP}_{n+1}}$ if and only if ${L}$ is homologically ${n}$-connected.
2. ${H_L\in \textit{FP}}$ if and only if ${L}$ is acyclic.
3. ${H_L}$ is finitely presented if and only if ${L}$ 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 ${n}$ we can provide a simple example of a group that is of type ${\textit{FP}_n}$ but not of type ${\textit{FP}_{n+1}}$. To see this, consider ${L=S^n}$, triangulated as an ${(n+1)}$-fold join of ${0}$-spheres. We get ${G_L=F_2\times F_2\times\ldots\times F_2}$ — that is, ${G_L}$ is the direct product of ${n+1}$ free groups on ${2}$ generators. Note that ${\widetilde{H}_n(L)=\mathbb{Z}}$ and ${\widetilde{H}_i(L)=0}$ for all ${i\neq n}$. Thus, from the Bestvina–Brady Theorem we can conclude that ${H_L}$ is of type ${\textit{FP}_n}$ since ${L}$ is homologically ${(n-1)}$-connected, but ${H_L}$ is not of type ${\textit{FP}_{n+1}}$ since ${L}$ is not homologically ${n}$-connected. In this example, the ${H_L}$ 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 ${L}$ of dimension ${2}$. One example of such a flag complex is a flag triangulation of a spine of the Poincaré homology sphere — the unique ${3}$-manifold with homology groups of a ${3}$-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 ${H_L}$ is of type ${\textit{FP}}$ (hence ${H_L}$ is of type ${\textit{FP}_2}$), but ${H_L}$ is not finitely presented.

Consider a presentation

$\displaystyle \Gamma=\langle a_1, \ldots, a_m|r_1,\ldots, r_n\rangle=F/R,$

where ${F}$ is free on the generators ${\{a_1, \ldots, a_m\}}$ and ${R=\langle\langle r_1,\ldots, r_n\rangle\rangle}$ — that is, ${R}$ is the normal closure of the subgroup generated by the elements ${\{r_1,\ldots, r_n\}}$. Note that ${F}$ acts on ${R}$ by conjugation, and hence induces an action of ${\Gamma}$ on ${R_{ab}:=\dfrac{R}{[R,R]}}$. On account of this action,

$\displaystyle \min \{ \, k \, | \, \exists s_1, \ldots, s_k\in F, R=\langle\langle s_1,\ldots, s_k\rangle\rangle\}$

is an upper bound on the rank of ${R_{ab}}$ as a ${\mathbb{Z}\Gamma}$-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 ${\Gamma=F/R}$, where ${F}$ is a free group of finite rank ${m}$. Then ${\Gamma}$ is of type ${\textup{FP}_2}$ iff ${R_{ab}}$ is finitely generated as a ${\mathbb{Z}\Gamma}$-module. A proof of this proposition based on Schanuel’s Lemma from homological algebra can be found in Michael Tweedale’s Thesis.

Now, let ${L}$ be an acyclic non-simply-connected finite flag complex of dimension ${2}$. As noted in the remark above, ${H_L}$ is of type ${\textit{FP}_2}$ but not of type ${F_2}$. Consequently, the rank of ${R_{ab}}$ as a ${\mathbb{Z}\Gamma}$-module is finite but ${H_L}$ is not finitely presented, leading to an infinite relation gap for ${H_L}$.

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, ${\textup{gd}(\Gamma)}$, of ${\Gamma}$ is the minimal ${n}$ such that there is a ${K(\Gamma, 1)}$ for ${\Gamma}$ of dimension ${n}$.

The cohomological dimension, ${\textup{cd}(\Gamma)}$, of ${\Gamma}$ is the minimal ${n}$ such that ${\mathbb{Z}}$ admits a resolution ${0\rightarrow P_n\rightarrow \ldots\rightarrow P_1\rightarrow P_0\rightarrow \mathbb{Z}\rightarrow 0}$ by projective ${\mathbb{Z}\Gamma}$-modules.

Note that for any group ${\Gamma}$, we have ${\textup{cd}(\Gamma)\leq \textup{gd}(\Gamma)}$ since the cellular chain complex of ${K(\Gamma, 1)}$ yields the desired projective resolution of ${\mathbb{Z}}$.

Theorem (Stallings–Swan). ${\textup{cd}(\Gamma)=1 \Leftrightarrow \textup{gd}(\Gamma)=1\Leftrightarrow \Gamma}$ is free.

Theorem (Eilenberg–Ganea). If ${\textup{cd}(\Gamma)\geq 3}$, then ${\textup{cd}(\Gamma)= \textup{gd}(\Gamma)}$.

So ${\textup{cd}(\Gamma)= \textup{gd}(\Gamma)}$ whenever ${\textup{cd}(\Gamma)\neq 2}$.

Conjecture (Eilenberg–Ganea). ${\textup{cd}(\Gamma)=2}$ implies ${\textup{gd}(\Gamma)=2}$.

Recall that a topological space ${X}$ 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 ${L}$ be the spine of the Poincaré Homology sphere as above. It is known that ${\textup{cd}(H_L)=2}$. Bestvina and Brady showed that if ${\textup{gd}(H_L)=2}$, as per the Eilenberg–Ganea Conjecture, then there exists a space contradicting the Whitehead Conjecture. Thus, one of these conjectures must be false.