  We show that every effectively closed action of a finitely generated group
$G$ on a closed subset of $\{0,1\}^{\mathbb{N}}$ can be obtained as a
topological factor of the $G$-subaction of a $(G \times H_1 \times
H_2)$-subshift of finite type (SFT) for any choice of infinite and finitely
generated groups $H_1,H_2$. As a consequence, we obtain that every group of the
form $G_1 \times G_2 \times G_3$ admits a non-empty strongly aperiodic SFT
subject to the condition that each $G_i$ is finitely generated and has
decidable word problem. As a corollary of this last result we prove the
existence of non-empty strongly aperiodic SFT in a large class of branch
groups, notably including the Grigorchuk group.

Dynamical Systems