A1 Refereed original research article in a scientific journal
Simulations and the lamplighter group
Authors: Bartholdi Laurent, Salo Ville
Publisher: EMS Press
Publication year: 2022
Journal: Groups, Geometry, and Dynamics
Journal name in source: GROUPS GEOMETRY AND DYNAMICS
Journal acronym: GROUP GEOM DYNAM
Volume: 16
Issue: 4
First page : 1461
Last page: 1514
Number of pages: 54
ISSN: 1661-7207
eISSN: 1661-7215
DOI: https://doi.org/10.4171/GGD/692(external)
Web address : https://ems.press/journals/ggd/articles/8202027(external)
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/178966923(external)
We introduce a notion of "simulation" for labelled graphs, in which edges of the simu-lated graph are realized by regular expressions in the simulating graph, and we prove that the tiling problem (a.k.a. the "domino problem") for the simulating graph is at least as difficult as that for the simulated graph. We apply this to the Cayley graph of the "lamplighter group" L = Z=2 Z Z, and more generally to "Diestel-Leader graphs". We prove that these graphs simulate the plane, and thus deduce that the seeded tiling problem is unsolvable on the group L. We note that L does not contain any plane in its Cayley graph, so our undecidability criterion by simulation covers cases not addressed by Jeandel's criterion based on translation-like action of a product of finitely generated infinite groups. Our approach to tiling problems is strongly based on categorical constructions in graph theory.
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