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<s}, 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

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About berstein

berstein is the name under which participants in the Berstein Seminar - a mathematics seminar at Cornell - are blogging.
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