Simulations and the lamplighter group



: Bartholdi Laurent, Salo Ville

Publisher: EMS Press

: 2022

: Groups, Geometry, and Dynamics

: GROUPS GEOMETRY AND DYNAMICS

: GROUP GEOM DYNAM

: 16

: 4

: 1461

: 1514

: 54

: 1661-7207

: 1661-7215

DOI: https://doi.org/10.4171/GGD/692

: https://ems.press/journals/ggd/articles/8202027

: https://research.utu.fi/converis/portal/detail/Publication/178966923


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.

Last updated on 2024-26-11 at 19:48