Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

We construct an example of a group G= Z²× G for a finite abelian group G, a subset E of G, and two finite subsets F₁, F₂ of G, such that it is undecidable in ZFC whether Z²× E can be tiled by translations of F₁, F₂. In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of E by the tiles F₁, F₂, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in Z²). A similar construction also applies for G= Z^d for sufficiently large d. If one allows the group G to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.