## Finiteness properties, Brown’s criterion and its application to Bieri-Stallings groups.

First we shall discuss an important result on finiteness properties of groups known as Brown’s Criterion. Then we shall use this result to prove the theorem discussed in our last post (and restated below) about finiteness properties of Bieri–Stallings groups.

Brown’s Criterion.

In order to state Brown’s Criterion, we need a few definitions.

Definitions. A directed set $D$ is a partially ordered set such that for each $\alpha,\beta \in D$ there is a $\gamma \in D$ with $\alpha\leq \gamma$ and $\beta\leq \gamma$. A directed system $(X_{\alpha})_{\alpha\in D}$ of objects in a category is a collection indexed by a directed set $D$ such that for every $\alpha,\beta \in D$ with $\alpha < \beta$, there is a morphism $f_{\alpha,\beta}:X_{\alpha}\rightarrow X_{\beta}$, and these morphisms satisfy

1. $f_{\alpha,\alpha}$ is an identity morphism, and
2. for each $\alpha<\beta<\gamma$ we have $f_{\alpha,\gamma}:X_{\alpha}\rightarrow X_{\gamma}=f_{\beta,\gamma}\circ f_{\alpha,\beta}$.

The direct limit of such a directed system $(X_{\alpha})_{\alpha\in D}$ is an object $X$ such that there are morphisms $g_{\alpha}:X_{\alpha}\rightarrow X$ which satisfy

1. if $\alpha < \beta$ then $g_{\alpha}=g_{\beta}\circ f_{\alpha,\beta}$; and
2. if there is another such object $Y$ with the maps $h_{\alpha}:X_{\alpha}\rightarrow Y$ satisfying (1), then there is a map $u:X\rightarrow Y$ such that $u\circ g_{\alpha}=h_{\alpha}$ for all $\alpha \in D$.

It can be shown that a direct limit, if it exists, is unique up to isomorphism in the strong sense that the isomorphism between the two candidates for the direct limit is exactly the map mentioned in (2).

A directed system of groups $(G_{\alpha})_{\alpha \in D}$ is essentially trivial if for any $\alpha \in D$ there is a $\beta \in D$ such that the map $G_{\alpha}\rightarrow G_{\beta}$ is trivial.

A directed system of CW complexes $(X_{\alpha})_{\alpha\in D}$ is essentially $n$-connected if $(\pi_n(X_{\alpha}))_{\alpha\in D}$ is essentially trivial.

Brown’s Criterion. Suppose $D$ is a directed set. Let $(X_{\alpha})_{\alpha \in D}$ be a directed system of CW complexes on which a group $G$ acts cocompactly by cell-permuting homeomorphisms. Suppose the direct limit $X$ is $(n-2)$-connected. If, for each cell $e$ of dimension less than or equal to $n$, the stabilizer $Stab_G(e)$ is of type $F_{n-dim(e)}$, then the following are equivalent.

1. The directed system $(\pi_i(X_{\alpha}))_{\alpha \in D}$ is essentially trivial for each $i.
2. $G$ is of type $F_n$

Finiteness properties of Bieri–Stallings groups

Recall that the Bieri–Stallings group $G_n$ is the kernel of the map

$\Phi_n:F(a_1,b_1)\times...\times F(a_n, b_n)\rightarrow \mathbb{Z}$

from the $n$-fold direct product of rank–two free groups to the integers which is defined by mapping all the $a_i$ and $b_i$ to $1$.

Theorem. $G_n$ is of type $F_{n-1}$, but not of type $F_n$.

Proof. We use a similar setup to that in our last post. Consider the map

$l:(S^1\vee S^1)\times...\times (S^1\vee S^1)\rightarrow S^1$,

where the domain is an $n$-fold product. The restriction of $l$ to each $S^1\vee S^1$ in the product is the map $m:S^1\vee S^1\rightarrow S^1$ that is the identity on each $S^1$, and $m$ extends “linearly” to $(S^1\vee S^1)\times...\times (S^1\vee S^1)$ — that is, $l(x_1,...,x_n)=\sum_{i=1}^nm(x_i)$. Note that $G_n$ is the kernel of the induced map

$l_{*}:\pi_1(S^1\vee S^1\times...\times S^1\vee S^1)\rightarrow \pi_1(S^1)$.

This map $l$ lifts to a Morse function between the covering spaces $f:T\times \cdots\times T\rightarrow \mathbb{R}$, where $T\times \cdots\times T$ is the $n$-fold product of the infinite $4$-valent tree $T$. The $n$-fold product ${ F(a_1,b_1) \times \cdots \times F(a_n,b_n)}$ acts on $T\times \cdots\times T$ in such as way that each ${F(a_i,b_i)}$ acts on the ${i}$-th factor $T$ in the product in the usual way.

We will first show that $G_n$ is of type $F_{n-1}$.

Consider the directed system $(X_t)_{t\in \mathbb{N}}$, where ${X_t=f^{-1}([-t,t])}$ and each map $g_{i,j}:X_i\rightarrow X_j$ for $j>i$ is the inclusion map. The direct limit is the product $T\times...\times T$, which is contractible and hence $n$-connected for each $n$. The action of $G_n$ on each $X_t$ is free, since this is just a restriction of a free action $F_2\times \cdots \times F_2$ acting on $T\times \cdots \times T$. This action is cocompact since the action restricted to each individual tree is cocompact. Also, it is clear that $G_n$ acts by cell permuting homeomorphisms. Note that each $X_t$ is invariant under the action of $G_n$, since the elements of this group act horizontally with respect to $f$.

Since the action is free, the stabilizer of any cell is trivial, and hence it satisfies the hypothesis of Brown’s criterion. We shall now show that the system $(\pi_i(X_t))_{t\in \mathbb N}$ is essentially trivial for $i.

First, we claim that the descending link as well as ascending link of any vertex is an $(n-1)$-sphere. The descending link of each vertex in each individual tree is the $0$ sphere $S^0$. The cone of a descending link of a vertex in the product of trees is the product of the individual cones of descending links in the individual trees, which are $0$-spheres. Since $CS^0\times \cdots \times CS^0=C(S^0 \ast \cdots \ast S^0)$, the descending link of any vertex in the product of trees is $(S^0 \ast \cdots\ast S^0)=S^{n-1}$.

By Morse’s Lemma, the only changes in topology that occur when extending the preimage from $X_t$ to the direct limit, are the coning off of ascending and descending links. Since these links are $(n-1)$-spheres and since the direct limit (a product of trees) is contractible, this implies that each $X_t$ must be $(n-2)$-connected. So $(\pi_i(X_t))_{t\in \mathbb N}$ is essentially trivial for $i. Therefore $G_n$ is of type $F_{n-1}$ by Brown's Criterion.

Now we will show that $G_n$ is not of type $F_n$.

By Brown’s criterion it suffices to show that the directed system mentioned above is not essentially $(n-1)$-connected. For a contradiction, let us assume that it is. Then for each $i$ there is a $j>i$ such that $\iota_{\star}:\pi_{n-1}(X_i)\rightarrow \pi_{n-1}(X_j)$, induced from the inclusion map, is trivial. We will show that there is a non trivial $(n-1)$-cycle in $f^{-1}(-j)$ that is homotopic in the product of trees to a non-trivial $(n-1)$-cycle in $f^{-1}(0)$, contrary to our assertion in the last sentence.

Consider the ascending link on a vertex of height $-j-1$, which is an $(n-1)$-sphere. This ascending link is the base of a cone with the vertex of the cone being our vertex. This cone is a product of cones in the individual trees. Each of these cones lie in larger cones inside each individual tree, whose product is a larger cone inside the product of trees which contains our cone (see illustration below).

Along a larger cone, we can push the $(n-1)$-cycle up to the level set $f^{-1}(0)$. This $(n-1)$-cycle is still non trivial in $X_j=f^{-1}([-j,j])$, since otherwise we would have a non trivial $n$-cycle in the contractible $n$-complex $T\times ...\times T$. Therefore $(\pi_{n-1}(X_t))_{t\in \mathbb{N}}$ is not essentially trivial, and hence $G_n$ is not of type $F_n$