## A sketch of a proof of the Bestvina–Brady Theorem

In this post we will sketch a proof of the Bestvina–Brady Theorem. Let us first recall the definition of the Bestvina–Brady group, which was introduced in the last post. Consider a flag complex ${L}$ and its associated right–angled Artin group ${G_L}$. Let ${\phi\colon G_L\rightarrow \mathbb{Z}}$ be the homomorphism sending each generator of ${G_L}$ to ${1\in\mathbb{Z}}$. We define the Bestvina–Brady group, ${H_L}$, to be the kernel of ${\phi}$.

Our goal is to relate the finiteness properties of ${H_L}$ to the topological properties of ${L}$. Before proceeding with the technical details, let us first give an informal overview of the argument. We will construct a ${K(G_L, 1)}$ space ${Q_L}$, and a map ${\psi\colon Q_L\rightarrow S^{1}}$, so that ${\phi\colon G_L\rightarrow \mathbb{Z}}$ is precisely the homomorphism induced by ${\psi}$ on the corresponding fundamental groups.

Lifting the map ${\psi}$ to the corresponding universal covers, we get a ${\phi}$-equivariant Morse function ${\widetilde{\psi}\colon\widetilde{Q}_L\rightarrow \mathbb{R}}$. Note that ${G_L}$ acts nicely by deck transformations on ${\widetilde{Q}_L}$. The key observation is that ${H_L}$ also acts nicely on the level sets of ${\widetilde{\psi}}$. We will analyze the topology of the level sets to get finiteness properties of ${H_L}$. The topology of the level sets ${f^{-1}(r)}$ can be recovered from sets ${f^{-1}(-\infty, r]}$ and ${f^{-1}[r,\infty)}$ via a Mayer–Vietoris sequence. These sets give information on ascending and descending links of vertices in ${\widetilde{Q}_L}$, which in turn give finiteness properties of ${H_L}$. We now begin the formal treatment.

For the rest of this post, we fix a finite flag complex ${L}$, the associated right-angled Artin group ${G_L}$, and the corresponding Bestvina–Brady group ${H_L\leq G_L}$. We now explicitly construct a ${K(G_L, 1)}$ space ${Q_L}$. The details of the construction will be important in what follows. Consider the Euclidean space ${\mathbb{R}^N}$, where ${N=|V(L)|}$, with each standard basis vector corresponding to a vertex of ${L}$. Let ${e_i\subset\mathbb{R}^N}$ be the line segment joining the origin and the ${i^{th}}$ basis vector. Now, for every simplex ${\sigma\in L}$, let ${\square_{\sigma}\subset\mathbb{R}^N}$ be the smallest cube containing the edges ${\{e_{i}:v_{i}\in \sigma\}}$. Finally, let ${Q_L}$ be the image of the set

$\displaystyle \bigcup \{\square_{\sigma}: \sigma \textup{ a simplex of } L\}$

under the projection ${\mathbb{R}^N\rightarrow T^N=\mathbb{R}^N/\mathbb{Z}^N}$.

Intuitively, ${Q_L}$ can be viewed as a CW–complex as follows. Its ${0}$–skeleton consists of a single point. The ${1}$–skeleton consists of a wedge of circles in one-to-one correspondence with the generators of ${G_L}$. The ${2}$-skeleton is obtained by attaching a ${2}$–torus by ${g_1g_2g_1^{-1}g_2^{-1}}$ for each edge ${\{g_1, g_2\}\in L}$. The ${3}$-skeleton is obtained by attaching a ${3}$–torus for each triangle in ${L}$, and so on. From this point of view, it is clear that ${Q_L}$ is a ${K(G_L,1)}$ space.

Now, consider a linear map ${l\colon\mathbb{R}^N\rightarrow \mathbb{R}}$, defined by ${(x_1, \ldots, x_N)\mapsto x_1+\ldots+x_N}$. It descends to a continuous map ${\mathbb{R}^N/\mathbb{Z}^N\rightarrow S^1}$. Restricting to ${Q_L}$, gives a map ${l\colon Q_L\rightarrow S^1}$.

Observation 1. The induced homomorphism on the fundamental groups, ${l_{*}\colon G_L\rightarrow \mathbb{Z}}$ sends each generator of ${G_L}$ to ${1\in\mathbb{Z}}$. Consequently, the homomorphism ${l_{*}}$ coincides with ${\phi}$.

Now consider the universal cover ${\widetilde{Q_L}}$ of ${Q_L}$ and the universal cover ${\mathbb{R}}$ of ${S^1}$. The groups ${G_L}$ and ${\mathbb{Z}}$ act as deck transformations on ${\widetilde{Q_L}}$ and ${\mathbb{R}}$, respectively. Lifting ${l}$ to the universal covers yields a continuous ${\phi}$-equivariant map ${f\colon\widetilde{Q_L}\rightarrow \mathbb{R}}$. To simplify notation, throughout the post we will denote ${\widetilde{Q_L}}$ by ${X}$.

Observation 2.${f}$ is a Morse function on ${X}$.

Proof: Since ${l}$ sends each edge of ${Q_L}$ homeomorphically onto ${S^1}$, we deduce that ${f}$ is non–constant on edges of ${X}$, and hence ${f}$ is also non–constant on higher dimensional cells. It is not difficult to check that ${f}$ is affine on every cell of ${X}$. Thus, ${f}$ is a Morse function on ${X}$. $\Box$

We omit the proof of the next observation, but the reasoning can be seen from the examples that follow. The proof can be found in Geoghegan’s Topologial Methods in Group Theory [Section 8.3].

Observation 3. The ${\uparrow}$-links and ${\downarrow}$-links of ${X}$ are homeomorphic to ${L}$.

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.

We will first prove the ${\Leftarrow"}$ implications for all three parts of the theorem, simultaneously. Then we will prove the ${\Rightarrow"}$ implications for the first two parts of the theorem and provide a sketch for the third part. For convenience, we will use ${H}$ to denote ${H_L}$.

Let us see how the topology of ascending and descending links gives information on finiteness properties of ${H}$. We will need the following lemmas that appeared in one of the previous posts. Here ${X_J:=f^{-1}(J)}$.

Lemma 4. Consider two nonempty, connected subsets ${J\subset J' \subset \mathbb{R}}$ such that the set ${X_{J'}\setminus X_J}$ contains no vertices of ${X}$. Then the inclusion ${X_J\hookrightarrow X_{J'}}$ is a homotopy equivalence.

Lemma 5 [Morse Lemma]. Let ${f\colon X\rightarrow \mathbb{R}}$ be a Morse function on ${X}$. Consider two closed intervals ${J\subset J' \subset \mathbb{R}}$ such that ${\inf J=\inf J'}$ and the set ${J'\setminus J}$ contains only one point ${t}$ of ${f(0\text{-cells})}$. Then ${X_{J'}}$ is homotopy equivalent to ${X_J}$ with copies of ${\text{Lk}_{\downarrow}(v,X)}$ coned off.

Corollary 6. Let ${f\colon X\rightarrow \mathbb{R}}$ be a Morse function and let ${J\subset J' \subset \mathbb{R}}$ be connected sets.

1. If each ${\uparrow}$-link and each ${\downarrow}$-link is homologically ${n}$-connected, then the inclusion ${X_J\hookrightarrow X_{J'}}$ induces an isomorphism on ${\widetilde{H}_i}$, for ${i\leq n}$, and an epimorphism on ${\widetilde{H}_{n+1}}$.
2. If each ${\uparrow}$-link and each ${\downarrow}$-link is simply-connected, then the inclusion ${X_J\hookrightarrow X_{J'}}$ induces an isomorphism on ${\pi_1}$.
3. If each ${\uparrow}$-link and each ${\downarrow}$-link is connected, then the inclusion ${X_J\hookrightarrow X_{J'}}$ induces an epimorphism on ${\pi_1}$.

Proof: First, observe that the set ${f^{-1}(X^{(0)})}$ is discrete. The proof now proceeds by induction using Lemma 4 and Lemma 5, as well as the Mayer-Vietoris and Seifert-Van Kampen Theorems, where the sets to be considered are ${X_J}$ and the cones on the descending/ascending links of the vertices ${v\in f^{-1}(J'\setminus J)}$. $\Box$

Using Corollary 6, we can now prove the following result.

Theorem 7. Let ${f\colon X\rightarrow \mathbb{R}}$ be a ${\phi}$-equivariant Morse function and let ${H}$ be the kernel of ${\phi}$.

1. If all ${\uparrow}$-links and ${\downarrow}$-links are homologically ${n}$-connected, then ${H\in \textup{FP}_{n+1}.}$
2. If all ${\uparrow}$-links and ${\downarrow}$-links are acyclic, then ${H\in \textup{FP}}$.
3. If all ${\uparrow}$-links and ${\downarrow}$-links are simply-connected, then ${H}$ is finitely presented.

Proof:

1. By Corollary 6, for any pair of real numbers ${t, the inclusion ${X_{(-\infty, t]}\hookrightarrow X_{(-\infty, s]}}$ induces an isomorphism on ${\widetilde{H}_i}$, for ${i\leq n}$, and an epimorphism on ${\widetilde{H}_{n+1}}$. Recall that ${X}$ is acyclic since it is a universal cover of a ${K(G_L,1)}$ space. Writing ${X}$ as ${X=\bigcup_{r\in \mathbb{Z}} X_{(-\infty, r]}}$, we deduce ${\widetilde{H}_i(X_{(-\infty, t]})=0}$ for each ${i\leq n}$ and all ${t}$. A similar argument yields ${\widetilde{H}_i(X_{[t,\infty)})=0}$ for each ${i\leq n}$ and all ${t}$. Now using the decomposition ${X=X_{(-\infty, t]}\cup X_{[t,\infty)}}$, it follows from the Mayer–Vietoris sequence that ${X_t=X_{(-\infty, t]}\cap X_{[t,\infty)}}$ is homologically ${n}$-connected. We now claim that the group ${H}$ is of type ${\textup{FP}_{n+1}}$. To see this, consider the cellular chain complex

$\displaystyle C_{n+1}(X_t)\rightarrow C_{n}(X_t)\rightarrow \ldots\rightarrow C_1(X_t)\rightarrow C_0(X_t)\rightarrow \mathbb{Z}\rightarrow 0.$

Since ${H}$ acts on ${X_t}$ freely and cocompactly, we deduce that each group ${C_i(X_t)}$ is finitely generated as a ${\mathbb{Z}H}$-module. Also observe that the chain complex is exact since ${X_t}$ is homologically ${n}$–connected. Thus, the chain gives rise to a partial resolution of ${\mathbb{Z}}$ by finitely generated ${\mathbb{Z}H}$–modules.

2. If all ${\uparrow}$-links and ${\downarrow}$-links are acyclic, we can proceed as above to show that ${X_t}$ is acyclic for any ${t}$. Observe that ${X}$ is finite dimensional. Hence, in the cellular chain complex above, ${C_i(X_t)}$ are non-zero for only finitely many ${i}$'s. Furthermore, since ${X_t}$ is acyclic, we deduce that this cellular chain is exact and hence we obtain a finite resolution of ${\mathbb{Z}}$ by finitely generated ${\mathbb{Z}H}$-modules.
3. If all ${\uparrow}$-links and ${\downarrow}$-links are simply-connected, then by Corollary 6, the inclusion ${X_t\hookrightarrow X}$ induces an isomorphism on ${\pi_1}$. So ${\pi_1(X_t)\cong \pi_1(X)\cong 1}$. Also, by part (1) of this theorem, ${X_t}$ is homologically ${1}$-connected and so ${X_t}$ is connected. Thus, ${X_t}$ is simply-connected. Now observe that the orbit space of ${X_t}$ can be made into a ${K(H,1)}$ space by adding cells of dimension ${3}$ and higher. Hence, ${H\in F_2}$.$\Box$

We are now ready for —

Proof of the ${\Leftarrow"}$ implication of the Bestvina–Brady Theorem:

1. If ${L}$ is homologically ${n}$-connected, then by Observation 3, ${\uparrow}$-links and ${\downarrow}$-links of ${X}$ are isomorphic to ${L}$. Now, part (1) of Theorem 7 implies ${H\in \textup{FP}_{n+1}}$.
2. If ${L}$ is acyclic, then by Observation 3, ${\uparrow}$-links and ${\downarrow}$-links of ${X}$ are acyclic. Now, part (2) of Theorem 7 implies ${H\in \textup{FP}}$.
3. If ${L}$ is simply-connected, then by Observation 3, ${\uparrow}$-links and ${\downarrow}$-links of ${X}$ are simply-connected. Now, part (3) of Theorem 7 implies that ${H}$ is finitely presented. $\Box$

We will now prove the ${\Rightarrow"}$ implication for the first statement of the Bestvina–Brady Theorem. We will need the following theorem, which we state here without proof.

Theorem 8. For each ${i}$, there is an isomorphism of ${\mathbb{Z}H}$-modules

$\displaystyle \widetilde{H}_{i}(X_t)\cong\oplus_{v\notin X_t}\widetilde{H}_{i}(L).$

Proof of the ${\Rightarrow"}$ implication of part (1) of the Bestvina–Brady Theorem:
Let ${H\in \textup{FP}_{n+1}}$. Set ${m=\min\{i:\widetilde{H}_i(L)\neq 0\}}$. Let us assume that ${m\leq n}$ and arrive at a contradiction. By Theorem 8, ${\widetilde{H}_{m}(X_t) \cong\oplus_{v\notin X_t}\widetilde{H}_{m}(L)}$. Since the Morse function, ${f}$, was onto ${\mathbb{Z}}$, we must have infinitely many vertices not in ${X_t}$. So, ${\widetilde{H}_{m}(X_t)}$ is not finitely generated as a ${\mathbb{Z}H}$-module. Consider the cellular chain complex

$\displaystyle C_m(X_t)\rightarrow C_{m-1}(X_t)\rightarrow \ldots\rightarrow C_1(X_t)\rightarrow C_0(X_t)\rightarrow \mathbb{Z}\rightarrow 0.$

Since ${H}$ acts cocompactly on ${X_t}$, we deduce that each ${C_i(X_t)}$ is a finitely generated free ${\mathbb{Z}H}$-module. Furthermore, from Theorem 8, we have ${\widetilde{H}_{i}(X_t)\cong\oplus_{v\notin X_t}\widetilde{H}_{i}(L)\cong 0}$ for all ${0\leq i\leq m-1}$, and hence the above chain is a partial projective resolution of finite type of length ${m}$. By theorem 4.3 in Brown's Cohomology of groups, ${K=\ker \{C_m(X_t)\rightarrow C_{m-1}(X_t)\}}$ is finitely generated (to apply the theorem, we use the fact that ${H\in \textup{FP}_{n+1}}$). But this implies that ${ \widetilde{H}_m(X_t)}$ is a quotient of ${K}$ and hence must be finitely generated, giving us the desired contradiction. Thus, ${L}$ is homologically ${n}$-connected. $\Box$

We can now easily deduce the forward direction of (2) in the Bestvina–Brady Theorem.

Proof of the ${\Rightarrow"}$ implication of part (2) of the Bestvina–Brady Theorem:
Suppose ${H\in \textup{FP}}$. Thus, there exists a finite resolution

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

by finitely generated projective ${\mathbb{Z}H}$-modules. Hence, for all ${m}$, there exists a partial resolution of ${\mathbb{Z}}$ of length ${m}$ by finitely generated projective ${\mathbb{Z}H}$-modules. So, for all ${m}$, we have ${H\in \textup{FP}_{m+1}}$ and therefore by part (1) of the Bestvina–Brady theorem, ${L}$ is homologically ${m}$-connected for all ${m}$. Thus, ${L}$ is acyclic. $\Box$

Proving the forward direction of the third part of the Bestvina–Brady Theorem requires much more work. Here is a sketch.

Proof of the ${\Rightarrow"}$ implication of part (3) of the Bestvina–Brady Theorem:
If ${L}$ is not connected, then ${L}$ is not homologically ${1}$-connected and by part (1) of the Bestvina–Brady Theorem we have ${H\notin \textup{FP}_2}$, and so ${H}$ is not finitely presented. Now if ${L}$ is connected but not simply–connected, we can show that ${\pi_1(X_t)}$ is generated by ${H}$–translates of finitely many loops. Since ${X}$ is contractible, the loops are null-homotopic in ${X}$, and so they must be null–homotopic in ${X_{[t-T, t+T]}}$ for some ${T}$. Now since ${H}$ acts by horizontal translations, ${H}$–orbits of loops are null-homotopic in ${X_{[t-T, t+T]}}$. Therefore, the inclusion ${X_t\hookrightarrow X_{[t-T, t+T]}}$ induces the trivial map on the fundamental groups. But it can be shown that this map must be an epimorphism on ${\pi_1}$'s and therefore ${X_{[t-T, t+T]}}$ must be simply–connected, which cannot happen if ${H}$ is finitely presented. $\Box$