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Is it true that a finitely generated subgroup of a cartesian product of free groups has a finite cohomological dimension?

The same question about pro-$p$ groups:

Is it true that a finitely generated closed subgroup of a cartesian product of free pro $p$-groups has a finite cohomological dimension?

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Regarding your first question, the answer is 'yes'. Consider an arbitrary direct product of free groups $\prod_\alpha F_\alpha$ and $H$ a finitely generated subgroup. Then $H$ is residually free. It follows from work of Baumslag--Remeslennikov--Miasnikov (I think, originally - there are now many proofs of this fact) that $H$ is a subgroup of a finite direct product of limit groups. Sela and Kharlampovich--Miasnikov proved that limit groups have finite $K(G,1)$'s, so the same is true of a finite direct product of limit groups. The covering space corresponding to $H$ is then a $K(H,1)$ with cells in only finitely many dimensions.

Let me add that it's not obvious how to prove the analogue of this in the pro-p world, as the pro-p analogues of limit groups are not well understood (or possibly even defined).

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