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

We realize the lamplighter group /2 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 Følner sets grow exponentially.