This still belongs to the framework of natural groups. The Lamplighter group as a wreath product or semi-direct product is a prototype group which illustrates some mathematics. First of all, the group, like the integers, is not a natural group. Given a metric structure invariant under the group, one can also find a different group structure which renders the group a Coxeter group and which is natural. There are relations to fundamental groups as some extension of the Lamplighter group is an example in the context of the open Atiyah conjecture of 1976 which in a strong form tells something about the Betti numbers of universal covers of manifolds with fundamental groups of bounded torsion. On the board there are two small issues to connect. First of all the direct sum should be over Z not N. The Lamplighter group is . The projection in the third part, one has the projection from all k-forms of the manifold to all harmonic forms. This story is actually getting things closer to some spectral theory I had been working on in my thesis. I still don’t know whether to get into this. Analytic things in the continuum are more technical. Still, the fractional Betti numbers are a challenge to compute. By the way, the discrete version of that Atiyah theory was already done one year after Atiyah and was already mentioned in Atiyah’s article. As expected, the story is exactly the same in the discrete and it shows also that the fractional Betti numbers do not depend on the metric of the Riemannian manifold.
