We introduce a notion of “simulation” for labelled graphs, in which edges of the simulated 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, 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.
1
Universität des Saarlandes, Saarbrücken, Germany
2
University of Turku, Finland
Laurent Bartholdi; Ville Salo. Simulations and the lamplighter group. Groups, geometry, and dynamics, Tome 16 (2022) no. 4, pp. 1461-1514. doi: 10.4171/ggd/692
@article{10_4171_ggd_692,
author = {Laurent Bartholdi and Ville Salo},
title = {Simulations and the lamplighter group},
journal = {Groups, geometry, and dynamics},
pages = {1461--1514},
year = {2022},
volume = {16},
number = {4},
doi = {10.4171/ggd/692},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/692/}
}
TY - JOUR
AU - Laurent Bartholdi
AU - Ville Salo
TI - Simulations and the lamplighter group
JO - Groups, geometry, and dynamics
PY - 2022
SP - 1461
EP - 1514
VL - 16
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/692/
DO - 10.4171/ggd/692
ID - 10_4171_ggd_692
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%0 Journal Article
%A Laurent Bartholdi
%A Ville Salo
%T Simulations and the lamplighter group
%J Groups, geometry, and dynamics
%D 2022
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%N 4
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/692/
%R 10.4171/ggd/692
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