Subgroups Of Direct Products Of Elementarily Free Groups

Abstract.

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.