### Abstract

SelaIf $\Gamma_1,\dots,\Gamma_n$ are limit groups and $S\subset\Gamma_1\times\dots\times\Gamma_n$ is of type ${\rm FP}_n(\mathbb Q)$ then $S$ contains a subgroup of finite index that is itself a direct product of at most $n$ limit groups. This answers a question of Sela.