Necessary conditions for tiling finitely generated amenable groups - Laboratoire Interdisciplinaire des Sciences du Numérique Access content directly
Journal Articles Discrete and Continuous Dynamical Systems - Series A Year : 2019

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 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.
Fichier principal
Vignette du fichier
Arxiv.pdf (389.73 Ko) Télécharger le fichier
Origin Files produced by the author(s)
Loading...

Dates and versions

hal-02092905 , version 1 (08-04-2019)

Identifiers

Cite

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⟩
154 View
103 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More