The Sigma Invariants of the Lamplighter Groups

New Algorithm For Computing Secondary Invariants of Invariant Rings of Monomial Groups

In this paper, a new  algorithm for computing secondary invariants of  invariant rings of monomial groups is presented. The main idea is to compute simultaneously a truncated SAGBI-G basis and the standard invariants of the ideal generated by the set of primary invariants.  The advantage of the presented algorithm lies in the fact that it is well-suited to complexity analysis and very easy to i...

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 Fractal Non-Contracting Class of Automata Groups

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 ...

Free Lamplighter Groups and a Question of Atiyah

We compute the von Neumann dimensions of the kernels of adjacency operators on free lamplighter groups and show that they are irrational, thus providing an elementary constructive answer to a question of Atiyah.