Subgroups of direct products of elementarily free groups

Abstract

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 $G_1,...,G_n$ are in E then a subgroup $\Gamma \subset G_1\times ...\times G_n$ is of type $\FP_n$ if and only if $\Gamma$ is itself, up to finite index, the direct product of at most $n$ groups from $\mathcal{E}$. This answers a question of Sela.

19 pages, no figures

References in corpus (1)

• Limit groups as limits of free groups: compactifying the set of free groups