Simulations and the Lamplighter group

TL;DR

This paper introduces a new notion of simulation for labelled graphs and demonstrates that certain groups, like the lamplighter group, have undecidable tiling problems by simulating the plane, expanding understanding of tiling problem complexities.

Contribution

It develops a novel graph simulation framework and applies it to prove the undecidability of the tiling problem for the lamplighter group and Diestel-Leader graphs.

Findings

Lamplighter group simulates the plane.

Undecidability of the tiling problem for the lamplighter group.

Simulation-based approach extends undecidability criteria.

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 prove that the tiling problem (aka "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" $L=\mathbb Z/2\wr\mathbb Z$, 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 covered 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.