Simulations and the lamplighter group

We introduce a notion of “simulation” for labelled graphs, in which edges the simulated graph are realized by regular expressions simulating graph, and we prove that tiling problem (a.k.a. “domino problem”) is at least as difficult graph. apply this to Cayley “lamplighter group” $L=\mathbb Z/2\wr\mathbb Z$, more generally “Diestel–Leader graphs”. these graphs simulate plane, thus deduce seeded unsolvable on group $L$. note $L$ does not contain any plane its so our undecidability criterion simulation covers cases addressed Jeandel’s based translation-like action product finitely generated infinite groups. Our approach problems strongly categorical constructions theory.