## Subgroups of products of surface and limit groups

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 ${G\leq \Gamma_1\times\cdots\times\Gamma_n}$, where ${\Gamma_i}$ are surface groups. If ${G}$ is of class ${\mbox{FP}_n}$, then it is virtually a direct product of at most ${n}$ surface groups.

Let ${G\leq \Gamma_1\times\cdots\times\Gamma_n}$ be a subgroup of any direct product. Let ${L_i = G\cap \Gamma_i}$. We say that ${G}$ is a subdirect product if all the ${L_i}$ are non-trivial. If ${\pi_i}$ is the projection from ${\Gamma_1\times\cdots\times\Gamma_n}$ that discards the ${i}$-th factor, then ${L_i=\ker(\pi_i|_G)}$. So if, say, ${L_n}$ is trivial, then ${\pi_n|_G}$ embeds ${G}$ into ${\Gamma_1\times\cdots\times\Gamma_{n-1}}$. 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 ${G\leq \Gamma_1\times\cdots\times\Gamma_n}$, where ${\Gamma_i}$ are surface groups. Assume ${L_i=G\cap\Gamma_i}$ are non-trivial, and arrange them so that ${L_1,\ldots,L_r}$ are not finitely generated and ${L_{r+1},\ldots,L_n}$ are finitely generated. Then there exists ${G_0 \leq G}$ of finite index such that

1. ${G_0=B\times L'_{r+1} \times \cdots \times L'_n}$ with ${L'_i\leq\Gamma_i}$ (thus a surface group), ${L'_i}$ finitely generated, and ${B\leq \Gamma_1\times\cdots\times\Gamma_r}$.
2. If ${r\geq 1}$, then ${H_r(B;{\mathbb Z})}$ is not finitely generated.

So ${G_0}$, which is of finite index in ${G}$, decomposes into a “nice” part ${L'_{r+1} \times \cdots \times L'_n}$ which verifies Theorem 1, and another factor ${B}$ which contains the obstructions to the conditions of Theorem 1.

If ${r=0}$, then ${G_0}$ is a product of at most ${n}$ surface groups. Moreover, these factors (${L'_j}$) are finitely generated and embedded in corresponding factors of the overgroup ${\Gamma_1\times\cdots\times\Gamma_n}$.

And if ${r\geq 1}$, then ${G}$ is not ${\mbox{FP}_r}$. To see this, let’s recall the definition of the homology groups ${H_*(G;{\mathbb Z})}$. They are obtained from a projective resolution of ${{\mathbb Z}}$ over ${{\mathbb Z} G}$

$\displaystyle \cdots \rightarrow P_r \rightarrow \cdots \rightarrow P_1 \rightarrow {\mathbb Z} G \rightarrow {\mathbb Z} \rightarrow 0$

by applying the tensor product ${-\otimes_{{\mathbb Z} G}{\mathbb Z}}$, and then taking the homology of the resulting chain complex. By this construction, if ${H_r(G;{\mathbb Z})}$ is not finitely generated then ${P_r}$ cannot be finitely generated either. Also, the homology groups satisfy Künneth formula for the direct product

$\displaystyle H_m(A\times B) = \bigoplus_{i=0}^m H_i(A)\otimes H_{n-i}(B)$

Hence, if ${H_r(B;{\mathbb Z})}$ is not finitely generated, then ${G_0}$ (and ${G}$) is not ${\mbox{FP}_r}$.

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 ${G\leq \Gamma_1\times\cdots\times\Gamma_n}$, where ${\Gamma_i}$ are limit groups. If ${G}$ is of class ${\mbox{FP}_n({\mathbb Q})}$, then it is virtually a direct product of at most ${n}$ limit groups.

This may be considered as a generalization of Theorem 1 in the case when the ${\Gamma_i}$ are finitely generated. A group ${G}$ is of class ${\mbox{FP}_n({\mathbb Q})}$ if ${{\mathbb Q}}$ admits a projective resolution over ${{\mathbb Q} G}$

$\displaystyle \cdots \rightarrow P_n \rightarrow \cdots \rightarrow P_1 \rightarrow {\mathbb Q} G \rightarrow {\mathbb Q} \rightarrow 0$

with ${P_j}$ finitely generated for ${j\leq n}$.

Theorem 4 (Theorem B, here). Let ${G\leq \Gamma_1\times\cdots\times\Gamma_n}$, where ${\Gamma_i}$ are limit groups. Assume ${L_i=G\cap\Gamma_i}$ are non-abelian, and arrange them so that ${L_1,\ldots,L_r}$ are not finitely generated and ${L_{r+1},\ldots,L_n}$ are finitely generated. Then there exists ${G_0 \leq G}$ of finite index such that

1. ${G_0=B\times L'_{r+1} \times \cdots \times L'_n}$ with ${L'_i\leq\Gamma_i}$, ${L'_i}$ finitely generated (so they are limit groups), and ${B\leq \Gamma_1\times\cdots\times\Gamma_r}$.
2. If ${r\geq 1}$, then ${\dim_{{\mathbb Q}} H_j(B;{\mathbb Q})=\infty}$ for some ${j\leq r}$.

Again, if ${r\geq 1}$ then ${G}$ is not of class ${\mbox{FP}_n({\mathbb Q})}$. So Theorem 3 reduces to 4. As an example of an application to residually free groups, we prove the following.

Corollary 5 (Corollary 1.1, here). If a finitely generated residually free group is of type ${\mbox{FP}_{\infty}({\mathbb Q})}$, then it is virtually a direct product of finitely many limit groups.

Proof: Let ${G}$ be a residually free group. As in the last post, ${G}$ embeds into a product ${\Gamma_1\times\cdots\times\Gamma_n}$ of limit groups. Now, since ${G}$ is ${\mbox{FP}_{\infty}({\mathbb Q})}$ (and in particular ${\mbox{FP}_n({\mathbb Q})}$), we can apply Theorem 3. This gives us the conclussion. $\Box$

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.

1. A residually free group is of type ${\mbox{F}_n}$ if and only if it is of type ${\mbox{FP}_n({\mathbb Q})}$.
2. A residually free group ${G}$ is finitely presented if and only if ${\dim H_2(G_0;{\mathbb Q})<\infty}$ for every ${G_0\leq G}$ of finite index.
3. The class of finitely presented residually free groups is recursively enumerable.
4. (${n}$-fold conjugacy problem) Let ${G}$ be a finitely presentable residually free group. There exists an algorithm that given two ${n}$-tuples ${(u_1,\ldots,u_n)}$ and ${(v_1,\ldots,v_n)}$ of words in the generators of ${G}$, determines whether or not there exists ${g\in G}$ such that ${u_i = gv_i g^{-1}}$ in ${G}$, for ${i=1,\ldots,n}$.
5. (Membership problem) Let ${G}$ be a finitely presentable residually free group and ${H\leq G}$ a finitely presentable subgroup. There exists an algorithm that given a word ${w}$ in the generators of ${G}$, determines whether or not ${w}$ represents an element in ${H}$.