Subgroups of the direct product of two finite rank free groups can be quite wild. In fact the direct product of two free groups contains very many subgroups, as per the following theorem of Baumslag and Roseblade.
Theorem 1. There are continuum many (i.e. the cardinality of ) non-isomorphic subgroups of .
While this theorem presents a daunting picture, it turns out that the finitely presented subgroups are much more tractable.
Theorem 2. Every finitely presented subgroup of a direct product of two free groups is either free or contains a finite index subgroup of the form where and .
This theorem has a long history. The original version was proved by Grunewald whose proof relied heavily on module theory. Next it was generalized by Baumslag and Roseblade using spectral sequences. Later, more geometric proofs were discovered: Short used van Kampen diagrams, while Bridson and Wise used a class of non-positively curved spaces. We will return to Bridson and Wise’s proof in our next post.
1. Mihailova’s Construction
Mihailova’s construction is the central idea behind both of the above theorems. It provides a means of associating a subgroup of of to any finitely generated group. Suppose a group is generated by , and let denote the free group on The Mihailova subgroup of is
To unpack this, let us assume we are given a presentation of – that is, a short exact sequence
in which is free. We will denote the map above by . In this case is the pullback group of the following diagram.
In the context of such a diagram, is often denoted by .
If has a finite presentation for , then is finitely generated. We can see that and form a generating set of because we can build any word in the in the first coordinate and and achieve any equivalent word in by a appending relators in the second coordinate. If we want to be finitely presented we are much more restricted:
Lemma 3 (Grunewald). Suppose is finitely presented and the normal closure of its relations in a finite rank free group is non-trivial. Then is finitely presented if and only if is finite.
However, is always recursively presented. Bogopolski and Ventura have shown that when is derived from a suitably nice presentation of , it has a one-parameter recursive presentation.
The “finite index” stipulation on Theorem 2 is in fact necessary. For instance, if then is not a non-trivial direct product (Bridson, Wise). This is essentially because the intersection of with each factor has index two in the factor, while is of index two in .
2. Applications of Mihailova’s construction
We close this post with some applications of Mihailova’s construction to the study of subgroups of . The first concerns the distortion of subgroups of .
The coincidence is due, in part, to Theorem 2 and a use of the Mihailova construction to obtain a finitely generated subgroup of from a finitely presented group.
One can also convert decidability problems from finitely generated groups into questions about subgroups of using Mihailova’s construction. The first such application was originally by Mihailova. We recall that the membership problem for a subgroup of a group concerns the determination of whether a word in the generators of G represents an element of H.
Theorem 5 (Mihailova). The membership problem for in is solvable if and only if the word problem for is solvable.
Since there are finitely presented groups with unsolvable word problem, this result implies that there are finitely generated subgroups of the direct product of two finite rank free groups with unsolvable membership problems.
Similar results can be obtained for other decidability problems. For instance, is orbit decidable if it can be determined whether if given there exists such that is conjugate to . Bogopolski, Martino, and Ventura have shown that undecidability of the word problem for implies that is isomorphic to an orbit undecidable subgroup of .