, Volume 17, Issue 2, pp 385-403
Date: 07 Mar 2007

Subgroups Of Direct Products Of Elementarily Free Groups

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The structure of groups having the same elementary theory as free groups is now known: they and their finitely generated subgroups form a prescribed subclass \({\mathcal{E}}\) of the hyperbolic limit groups. We prove that if G 1,...,G n are in \({\mathcal{E}}\) then a subgroup Γ ⊂ G 1 × … × G n is of type FP n if and only if Γ is itself, up to finite index, the direct product of at most n groups from \({\mathcal{E}}\) . This provides a partial answer to a question of Sela.

This work was supported in part by Franco–British Alliance project PN 05.004. The first author is also supported by an EPSRC Senior Fellowship and a Royal Society Wolfson Research Merit Award.
Received: July 2005 Accepted: April 2006