The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum*

Abstract

We realize the lamplighter group \(\mathbb{Z}\)/2\(\mathbb{Z}\) ≀ \(\mathbb{Z}\) as a group defined by a 2-state automaton. We study the corresponding action of this group on a binary tree and on its boundary. The final goal is the computation for a special system of generators of the spectrum of the Markov (or the random walk) operator which is [−1,1] in this case and of the spectral measure which is a discrete measure concentrated on a dense countable set of points in [−1,1] (a new effect unseen before for Markovian operators on groups which leads to a counterexample to the Strong Atiyah Conjecture). This is done by the computation of spectra of finite-dimensional approximations of the operator and uses an idea of fractalness in a similar way it was used by Bartholdi and Grigorchuk for the computation of the spectra of some branch groups. We also obtain the asymptotic of type e−1/1−x of the spectral measure in the neighborhood of 1 and show that Folner sets grow exponentially.