Finitely generated subgroups of a product of free groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:06:47Z http://mathoverflow.net/feeds/question/58410 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58410/finitely-generated-subgroups-of-a-product-of-free-groups Finitely generated subgroups of a product of free groups Andrei Jaikin 2011-03-14T10:48:23Z 2011-03-14T15:12:26Z <p>Is it true that a finitely generated subgroup of a cartesian product of free groups has a finite cohomological dimension?</p> <p>The same question about pro-$p$ groups:</p> <p>Is it true that a finitely generated closed subgroup of a cartesian product of free pro $p$-groups has a finite cohomological dimension?</p> http://mathoverflow.net/questions/58410/finitely-generated-subgroups-of-a-product-of-free-groups/58433#58433 Answer by HW for Finitely generated subgroups of a product of free groups HW 2011-03-14T15:01:08Z 2011-03-14T15:12:26Z <p>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.</p> <p>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).</p>