We show that the Plancherel measure of the lamplighter random walk on a graph coincides with the expected spectral measure of the absorbing random walk on the Bernoulli percolation clusters. In the subcritical regime the spectrum is pure point and we construct a complete set of finitely supported eigenfunctions.

We derive Donsker-Vardhan type results for functionals of the occupation times when the underlying random walk on Z is in the domain of attraction of an operatorstable law on R. Applications to random walks on wreath products (also known as lamplighter groups) are given.

For a restricted wreath product $G\wr \mathbb{Z}^k$, where $G$ is finite abelian group, we determine (almost in all cases) whether this has the $R_\infty$ property (i.e., each its automorphism infinite Reidemeister number).

Main theorems of the article concern the problem of M. Atiyah on possible values of l2-Betti numbers. It is shown that all non-negative real numbers are l2-Betti numbers, and that “many” (for example all non-negative algebraic) real numbers are l2-Betti numbers of aspherical manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to an aspherical m...

Abstract From any two median spaces $X$ and $Y$ , we construct a new space $X \circledast Y$ referred to as the diadem product of show that this construction is compatible with wreath products in following sense: given finitely generated groups $G,\,H$ (equivariant) coarse embeddings into $X,\,Y$ there exist a(n equivariant) embedding $G\wr H \to X . The offers unified point view on various que...