This post is about strong generalizations of Baumslag and Roseblade’s theorem about subgroups of direct products of free groups. These generalizations were obtained by Bridson, Howie, Miller and Short. First they consider subgroups of direct products of surface groups. In this context, surface groups will be fundamental groups of orientable surfaces, which may have boundary, and may or may not be compact. Free groups are examples of surface groups. So the next theorem generalizes Baumslag and Roseblade’s.
Theorem 1 (Theorem A, here). Let , where are surface groups. If is of class , then it is virtually a direct product of at most surface groups.
Let be a subgroup of any direct product. Let . We say that is a subdirect product if all the are non-trivial. If is the projection from that discards the -th factor, then . So if, say, is trivial, then embeds into . This allows us to reduce the above theorem to subdirect products. This also allows for a more specific result.
Theorem 2 (Theorem B, here). Let , where are surface groups. Assume are non-trivial, and arrange them so that are not finitely generated and are finitely generated. Then there exists of finite index such that
- with (thus a surface group), finitely generated, and .
- If , then is not finitely generated.
If , then is a product of at most surface groups. Moreover, these factors () are finitely generated and embedded in corresponding factors of the overgroup .
And if , then is not . To see this, let’s recall the definition of the homology groups . They are obtained from a projective resolution of over
by applying the tensor product , and then taking the homology of the resulting chain complex. By this construction, if is not finitely generated then cannot be finitely generated either. Also, the homology groups satisfy Künneth formula for the direct product
Hence, if is not finitely generated, then (and ) is not .
Next we consider analogous results on subgroups of products of limit groups. In the last post we saw that finitely generated residually free groups are exactly the groups that embed into direct products of limit groups. Thus, such theorems will allow us to study the class of residually free groups.
Theorem 3 (Theorem A, here). Let , where are limit groups. If is of class , then it is virtually a direct product of at most limit groups.
This may be considered as a generalization of Theorem 1 in the case when the are finitely generated. A group is of class if admits a projective resolution over
with finitely generated for .
Theorem 4 (Theorem B, here). Let , where are limit groups. Assume are non-abelian, and arrange them so that are not finitely generated and are finitely generated. Then there exists of finite index such that
- with , finitely generated (so they are limit groups), and .
- If , then for some .
Corollary 5 (Corollary 1.1, here). If a finitely generated residually free group is of type , then it is virtually a direct product of finitely many limit groups.
Proof: Let be a residually free group. As in the last post, embeds into a product of limit groups. Now, since is (and in particular ), we can apply Theorem 3. This gives us the conclussion.
In their third paper, Bridson, Howie, Miller and Short were able to use the above theorems in order to prove many results about residually free groups. Here is a sample.
- A residually free group is of type if and only if it is of type .
- A residually free group is finitely presented if and only if for every of finite index.
- The class of finitely presented residually free groups is recursively enumerable.
- (-fold conjugacy problem) Let be a finitely presentable residually free group. There exists an algorithm that given two -tuples and of words in the generators of , determines whether or not there exists such that in , for .
- (Membership problem) Let be a finitely presentable residually free group and a finitely presentable subgroup. There exists an algorithm that given a word in the generators of , determines whether or not represents an element in .