Necessary conditions for tiling finitely generated amenable groups

Benjamin Hellouin de Menibus (1) , Hugo Maturana Cornejo (2)

1 GALaC - Graphes, Algorithmes et Combinatoire - LISN
2 Departamento de Ingenieria Matematica, Centro Modelamiento Matematico

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.

Mots clés: domino problem; symbolic dynamics; tilings; groups; periodicity; amenability

Domaines: Systèmes dynamiques [math.DS]; Théorie et langage formel [cs.FL]; Théorie des groupes [math.GR]

Benjamin Hellouin de Menibus, Hugo Maturana Cornejo. Necessary conditions for tiling finitely generated amenable groups. Discrete and Continuous Dynamical Systems - Series A, 2019, 40 (4), pp.2335-2346. ⟨10.3934/dcds.2020116⟩. ⟨hal-02092905⟩