## Subgroups of the direct product of two free groups

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 $\mathbb{R}$) non-isomorphic subgroups of ${F_2\times F_2}$.

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 ${F\times F'}$ is either free or contains a finite index subgroup of the form ${H\times H'}$ where ${H\leq F}$ and ${H'\leq F'}$.

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 ${F_n\times F_n}$ to any finitely generated group. Suppose a group ${G}$ is generated by ${\{a_1,..,a_n\}}$, and let ${F_n}$ denote the free group on ${ a_1,..,a_n. }$ The Mihailova subgroup of ${F_n\times F_n}$ is

$\displaystyle M(G)=\{(w_1,w_2)\in F_n\times F_n | w_1 =w_2 \textrm{ in } G\}. \ \ \ \ \ (1)$

To unpack this, let us assume we are given a presentation of ${G}$ – that is, a short exact sequence

$\displaystyle 1 \rightarrow R \rightarrow F \rightarrow G \rightarrow 1 \ \ \ \ \ (2)$

in which $F$ is free. We will denote the map ${F\rightarrow G}$ above by ${\phi}$. In this case ${M(G)}$ is the pullback group of the following diagram.

${\begin{array}{clcccr} M(G) & \rightarrow & F \\ \downarrow & & \downarrow\phi \\ F & \overrightarrow{\phi} & G\\ \end{array}}$

In the context of such a diagram, $M(G)$ is often denoted by ${F\times_\phi F}$.

If $G$ has a finite presentation ${}$ for ${G}$, then ${M(G)}$ is finitely generated. We can see that ${(a_1,a_1),...,(a_n,a_n)}$ and ${(1,r_1),...,(1,r_m)}$ form a generating set of ${M(G)}$ because we can build any word in the $a_i$ in the first coordinate and and achieve any equivalent word in $G$ by a appending relators in the second coordinate. If we want ${M(G)}$ to be finitely presented we are much more restricted:

Lemma 3 (Grunewald). Suppose ${G}$ is finitely presented and the normal closure of its relations in a finite rank free group ${F}$ is non-trivial. Then ${M(G)}$ is finitely presented if and only if ${G}$ is finite.

However, ${M(G)}$ is always recursively presented. Bogopolski and Ventura have shown that when ${M(G)}$ is derived from a suitably nice presentation of ${G}$, it has a one-parameter recursive presentation.

The “finite index” stipulation on Theorem 2 is in fact necessary. For instance, if ${\phi:F\rightarrow {\mathbb Z}_2}$ then ${M({\mathbb Z}_2)}$ is not a non-trivial direct product (Bridson, Wise). This is essentially because the intersection of ${M({\mathbb Z}_2)}$ with each factor has index two in the factor, while ${M({\mathbb Z}_2)}$ is of index two in ${F\times F}$.

2. Applications of Mihailova’s construction

We close this post with some applications of Mihailova’s construction to the study of subgroups of ${F\times F}$. The first concerns the distortion of subgroups of ${F\times F}$.

Theorem 4 (Olshanskii, Sapir). The Dehn functions for finitely presented groups coincides up to ${\simeq}$equivalence with the distortion functions for finitely-generated subgroups of ${F\times F}$.

The coincidence is due, in part, to Theorem 2 and a use of the Mihailova construction to obtain a finitely generated subgroup of ${F\times F}$ from a finitely presented group.

One can also convert decidability problems from finitely generated groups into questions about subgroups of ${F\times F}$ using Mihailova’s construction. The first such application was originally by Mihailova. We recall that the membership problem for a subgroup ${H}$ of a group ${G}$ concerns the determination of whether a word in the generators of G represents an element of H.

Theorem 5 (Mihailova). The membership problem for ${M(G)}$ in ${F_n\times F_n}$ is solvable if and only if the word problem for ${G}$ 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, ${A is orbit decidable if it can be determined whether if given ${u,v\in F}$ there exists ${a\in A}$ such that ${u}$ is conjugate to ${a\cdot v}$. Bogopolski, Martino, and Ventura have shown that undecidability of the word problem for ${G}$ implies that ${M(G)}$ is isomorphic to an orbit undecidable subgroup of ${Aut(F_3)}$.