On a Class of Automata Groups Generalizing Lamplighter Groups

A Fractal Non-Contracting Class of Automata Groups

Cross-wired lamplighter groups and linearity of automata groups

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...

A note on the Poisson boundary of lamplighter random walks

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups Γ endowed with a “rich” boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich [16]. A geometrical method for constructing the strip as a subset of the lamplighter group ...

Some Solvable Automaton Groups

It is shown that certain ascending HNN extensions of free abelian groups of finite rank, as well as various lamplighter groups, can be realized as automaton groups, i.e., can be given a self-similar structure. This includes the solvable Baumslag-Solitar groups BS(1, m), for m 6= ±1. In addition, it is shown that, for any relatively prime integers m, n ≥ 2, the pair of Baumslag-Solitar groups BS...

The Spectra of Lamplighter Groups and Cayley Machines

We calculate the spectra and spectral measures associated to random walks on restricted wreath products G wr Z, with G a finite abelian group, by realizing them as groups generated by automata. This generalizes the work of Grigorchuk and Żuk on the lamplighter group. More generally we calculate the spectra of random walks on groups generated by Cayley machines of finite groups and calculate Kes...