2 3 Ju n 20 05 Subgroups of direct products of elementarily free groups

We exploit Zlil Sela’s description of the structure of groups having the same elementary theory as free groups: they and their finitely generated subgroups form a prescribed subclass E of the hyperbolic limit groups. We prove that if G1, . . . , Gn are in E then a subgroup Γ ⊂ G1 × · · · × Gn is of type FPn if and only if Γ is itself, up to finite index, the direct product of at most n groups from E . This answers a question of Sela. Examples of Stallings [24] and Bieri [4] show that finitely presented subgroups of a direct product of finitely many free groups can be rather wild. In contrast, Baumslag and Roseblade [1] proved that the only finitely presented subgroups of a direct product of two free groups are the obvious ones: such a subgroup is either free or else has a subgroup of finite index that is the product of its intersections with the factors. In [9] Miller, Short and the present authors explained this apparent contrast by showing that all of the exotic behaviour among subdirect products of free groups arises from a lack of homological finiteness. More precisely, if a subgroup S of a direct product of n free groups has finitely generated homology up to dimension n, then S has a subgroup of finite index that is isomorphic to a direct product of free groups. We proved a similar theorem for subdirect products of surface groups. This has implications for the understanding of the fundamental groups of compact Kähler manifolds. Indeed the remarkable work of Delzant and Gromov [11] shows that if the fundamental group Γ of such a manifold is torsion-free and has sufficiently many multi-ended splittings, then there is a short exact sequence 1 → Z → Γ0 → S → 1, where S is a subdirect product of surface groups and Γ0 is a subgroup of finite index in Γ. Different attributes of surface groups enter the proof in [9] in subtle ways making it difficult to assess which are vital and which are artifacts of the proof. Examples show that the splitting phenomenon for subdirect products does not extend to arbitrary 2-dimensional hyperbolic groups (even small cancellation groups), nor to fundamental