Undecidability of translational monotilings

In the 60’s, Berger famously showed that translational tilings of ${\mathbb{Z}}^2$ with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability of translational monotilings (tilings by translations of a single tile) in ${\mathbb{Z}}^2$. The decidability of translational monotilings in higher dimensions remained unsolved. In this paper, by combining our recently developed techniques with ideas introduced by Aanderaa and Lewis, we finally settle this problem, achieving the undecidability of translational monotilings of (periodic subsets of) virtually ${\mathbb{Z}}^2$ spaces, namely, spaces of the form ${\mathbb{Z}}^2\times G_0$, where $G_0$ is a finite Abelian group. This also implies the undecidability of translational monotilings in ${\mathbb{Z}}^d$, $d\ge 3$.