Applications of the Bestvina-Brady Theorem

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}.

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, {\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.


About berstein

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