Necessary conditions for tiling finitely generated amenable groups

Abstract

We consider a set of necessary conditions which are efficient heuristics for deciding when a set of Wang tiles cannot tile a group.

Piantadosi [19] gave a necessary and sufficient condition for the existence of a valid tiling of any free group. This condition is actually necessary for the existence of a valid tiling for an arbitrary finitely generated group.

We consider two other conditions: the first, also given by Piantadosi [19], is a necessary and sufficient condition to decide if a set of Wang tiles gives a strongly periodic tiling of the free group; the second, given by Chazottes et. al. [9], is a necessary condition to decide if a set of Wang tiles gives a tiling of Z(2).

We show that these last two conditions are equivalent. Joining and generalising approaches from both sides, we prove that they are necessary for having a valid tiling of any finitely generated amenable group, confirming a remark of Jeandel [14]