Simulations and the lamplighter group

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 D Z=2 o 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.

Strongly aperiodic SFTs on generalized Baumslag–Solitar groups

“…The commonly accepted conjecture states that virtually free groups are precisely those with decidable domino problem. We refer to [2, 11] for recent advances on this problem. In this article, we will only make use of the fact that the domino problem is undecidable on and that if H is a subgroup of G , then if H has undecidable domino problem, then so has G [1].…”

Monadic second-order logic and the domino problem on self-similar graphs

“…In [3], together with Ville Salo, I consider the domino problem on a Cayley graph which does not contain any grid, the "lamplighter group" Z=2 o Z, and show that nevertheless its "seeded" domino problem (see Section 2.1) is undecidable. The main, general idea is that an auxiliary domino problem may be used to mark some vertices and some sequences of edges to simulate a grid within .…”

“…(In fact, it would be equally good to simulate any graph with unsolvable domino problem, but somehow we always fall back on the grid.) This is the argument used in Section 5 to prove Theorems B and C; though we do not make use of the general results of [3], and rather repeat the argument in each specific case.…”

“…Simulations. We recall in simplified form the definition of simulations from [4]. We consider vertex-and edge-colourings specified on an edge-labeled graph Γ, and say that Γ simulates a graph ∆ if there is a set of domino tiles for Γ whose solutions have the following properties:…”

“…The set of principal outgoing edges defines a spanning tree Υ of ΓpG, Sq, and }g ´1} Υ " }g ´1} ΓpG,Sq " }g} for all g P G. Furthermore, Υ is produced by a regular grammar [13,Theorem 3,4,5]: every vertex has one of finitely many types; and the type of a vertex determines the types of all of its descendants. (In this manner, Cannon [9] proves that the growth series of G is a rational function.)…”