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Necessary conditions for tiling finitely generated amenable groups

  • * Corresponding author: Benjamin Hellouin de Menibus

    * Corresponding author: Benjamin Hellouin de Menibus 

This article was written during stays of the first author funded by an LRI internal project. The second author was partially funded by the ECOS-SUD project C17E08, the ANR project CoCoGro (ANR-16-CE40-0005) and CONICYT doctoral fellowship 21170770

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  • 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 $ \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 [14].

    Mathematics Subject Classification: Primary: 37B50; Secondary: 37B10, 05B45.


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  • Figure 1.  Examples of Wang tiles with colours $ \mathcal{C} = \left\{ {a,b,c,d} \right\}$ on one and two generators, respectively, with their corresponding maps

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