We consider the two generalizations of lamplighter groups: automata groups generated by Cayley machines and cross-wired lamplighter groups. For a finite step two nilpotent group with central squares, we study its associated Cayley machine and give a presentation of the corresponding automata group. We show the automata group is a crosswired lamplighter group and does not embed in the wreath pro...

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 ...

We study languages of geodesics in lamplighter groups and Thompson’s group F . We show that the lamplighter groups Ln have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group ...

We study languages of geodesics in lamplighter groups and Thompson’s group F . We show that the lamplighter groups Ln have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group ...

Let G be a finitely generated group and X its Cayley graph with respect to a finite, symmetric generating set S. Furthermore, let H be a finite group and H ≀G the lamplighter group (wreath product) over G with group of “lamps” H. We show that the spectral measure (Plancherel measure) of any symmetric “switch–walk–switch” random walk on H ≀G coincides with the expected spectral measure (integrat...

Suppose we are given an infinite, finitely generated group G and a transient random walk with bounded range on the wreath product (Z/2Z) ≀ G, such that its projection on G is transient. This random walk can be interpreted as a lamplighter random walk, where there is a lamp at each element of G, which can be switched on and off, and a lamplighter walks along G and switches lamps randomly on and ...