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.
In order to state Brown’s Criterion, we need a few definitions.
Definitions. A directed set is a partially ordered set such that for each there is a with and . A directed system of objects in a category is a collection indexed by a directed set such that for every with , there is a morphism , and these morphisms satisfy
- is an identity morphism, and
- for each we have .
The direct limit of such a directed system is an object such that there are morphisms which satisfy
- if then ; and
- if there is another such object with the maps satisfying (1), then there is a map such that for all .
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 is essentially trivial if for any there is a such that the map is trivial.
A directed system of CW complexes is essentially -connected if is essentially trivial.
Brown’s Criterion. Suppose is a directed set. Let be a directed system of CW complexes on which a group acts cocompactly by cell-permuting homeomorphisms. Suppose the direct limit is -connected. If, for each cell of dimension less than or equal to , the stabilizer is of type , then the following are equivalent.
- The directed system is essentially trivial for each .
- is of type
Finiteness properties of Bieri–Stallings groups
Recall that the Bieri–Stallings group is the kernel of the map
from the -fold direct product of rank–two free groups to the integers which is defined by mapping all the and to .
Theorem. is of type , but not of type .
Proof. We use a similar setup to that in our last post. Consider the map
where the domain is an -fold product. The restriction of to each in the product is the map that is the identity on each , and extends “linearly” to — that is, . Note that is the kernel of the induced map
This map lifts to a Morse function between the covering spaces , where is the -fold product of the infinite -valent tree . The -fold product acts on in such as way that each acts on the -th factor in the product in the usual way.
We will first show that is of type .
Consider the directed system , where and each map for is the inclusion map. The direct limit is the product , which is contractible and hence -connected for each . The action of on each is free, since this is just a restriction of a free action acting on . This action is cocompact since the action restricted to each individual tree is cocompact. Also, it is clear that acts by cell permuting homeomorphisms. Note that each is invariant under the action of , since the elements of this group act horizontally with respect to .
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 is essentially trivial for .
First, we claim that the descending link as well as ascending link of any vertex is an -sphere. The descending link of each vertex in each individual tree is the sphere . 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 -spheres. Since , the descending link of any vertex in the product of trees is .
By Morse’s Lemma, the only changes in topology that occur when extending the preimage from to the direct limit, are the coning off of ascending and descending links. Since these links are -spheres and since the direct limit (a product of trees) is contractible, this implies that each must be -connected. So is essentially trivial for . Therefore is of type by Brown's Criterion.
Now we will show that is not of type .
By Brown’s criterion it suffices to show that the directed system mentioned above is not essentially -connected. For a contradiction, let us assume that it is. Then for each there is a such that , induced from the inclusion map, is trivial. We will show that there is a non trivial -cycle in that is homotopic in the product of trees to a non-trivial -cycle in , contrary to our assertion in the last sentence.
Consider the ascending link on a vertex of height , which is an -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 -cycle up to the level set . This -cycle is still non trivial in , since otherwise we would have a non trivial -cycle in the contractible -complex . Therefore is not essentially trivial, and hence is not of type