Sandpile monomorphisms and limits

Moritz Lang; Mikhail Shkolnikov

doi:10.5802/crmath.291

Algèbre, Théorie des groupes

Sandpile monomorphisms and limits

Moritz Lang ¹ ; Mikhail Shkolnikov ²

¹ University of Applied Sciences Technikum Wien, Höchstädtplatz 6, 1200 Wien, Austria
² Université de Genève, Section de mathématiques, route de Drize 7, villa Battelle, 1227 Carouge, Switzerland

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 333-341.

Résumé

We introduce a tiling problem between bounded open convex polyforms $\hat{P} \subset ℝ^{2}$ with colored directed edges. If there exists a tiling of the polyform ${\hat{P}}_{2}$ by ${\hat{P}}_{1}$ , we construct a monomorphism from the sandpile group $G_{Γ_{1}} = ℤ^{Γ_{1}} / Δ (ℤ^{Γ_{1}})$ on $Γ_{1} = {\hat{P}}_{1} \cap ℤ^{2}$ to the one on $Γ_{2} = {\hat{P}}_{2} \cap ℤ^{2}$ . We provide several examples of infinite series of such tilings converging to $ℝ^{2}$ , and thus define the limit of the sandpile group on the plane.

Reçu le : 2021-01-01
Révisé le : 2021-08-25
Accepté le : 2021-10-24
Publié le : 2022-04-26

DOI : 10.5802/crmath.291

Classification : 20K30, 60K35, 47D07

Affiliations des auteurs :

Moritz Lang ¹ ; Mikhail Shkolnikov ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Moritz Lang and Mikhail Shkolnikov},
     title = {Sandpile monomorphisms and limits},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {333--341},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.291},
     language = {en},
}

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Moritz Lang; Mikhail Shkolnikov. Sandpile monomorphisms and limits. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 333-341. doi : 10.5802/crmath.291. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.291/

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