# Necessary conditions for tiling finitely generated amenable groups

## TL;DR

This paper explores necessary conditions for tiling finitely generated amenable groups with Wang tiles, extending known conditions from free groups and a5^2 to a broader class of groups, and establishing their equivalence.

## Contribution

It generalizes and unifies necessary tiling conditions from free groups and a5^2 to all finitely generated amenable groups, confirming a conjecture.

## Key findings

- Necessary conditions are equivalent for tilings of free groups and a5^2.
- These conditions are necessary for tilings of any finitely generated amenable group.
- The paper confirms a conjecture by Jeandel regarding tiling conditions.

## 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 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 then consider two other conditions: the first, also given by Piantadosi, 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., is a necessary condition to decide if a set of Wang tiles gives a tiling of $\mathbb Z^2$. 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.