# Simulations and the lamplighter group

### Laurent Bartholdi

Universität des Saarlandes, Saarbrücken, Germany### Ville Salo

University of Turku, Finland

## Abstract

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.

## Supplementary Material

## Cite this article

Laurent Bartholdi, Ville Salo, Simulations and the lamplighter group. Groups Geom. Dyn. 16 (2022), no. 4, pp. 1461–1514

DOI 10.4171/GGD/692