Continuous graph homomorphisms of higher dimensional abelian group actions

For every fixed integer $d\ge 2$, we prove that the finite graphs receiving a continuous homomorphism from the standard Schreier graph $F(2^{{\mathbb{Z}}^d})$ form a ${\mathrm{\Sigma }}_1^0$-complete set. This extends a theorem of Gao, Jackson, Krohne and Seward when $d=2$. For the positive part of the reduction, we show that if a graph $H$ satisfies a certain property then there is a continuous graph homomorphism from $F(2^{{\mathbb{Z}}^d})$ to $H$, in particular, the complete graph $K_4$ satisfies this property. This extends a theorem of Gao and Jackson. We also prove that $\left({\mathcal{H}}_d\right)$ is a strictly decreasing sequence.