We settle an open problem, raised by Y. Peres and D. Revelle, concerning the L mixing time of the random walk on the lamplighter graph. We also provide general bounds relating the entropy decay of a Markov chain to the separation distance of the chain, and show that the lamplighter graphs once again provide examples of tightness of our results.

We study automata groups generated by reset automata. Every lamplighter group Z/nZ wr Z can be generated by such an automaton, and in general these automata groups are similar in nature to lamplighters: they are amenable locally-finite-by-cyclic groups; under mild decidable conditions, the semigroups generated by such automata are free. Parabolic subgroups and fractal properties are considered.

We answer positively a question of Kaimanovich and Vershik from 1979, showing that the final configuration of lamps for simple random walk on the lamplighter group over Z (d ≥ 3) is the Poisson boundary. For d ≥ 5, this had been shown earlier by Erschler (2011). We extend this to walks of more general types on more general groups.

We show that the lamplighter group L has a system of generators for which spectrum discrete Laplacian on Cayley graph is union an interval and countable set isolated points accumulating to point outside this interval. This first example with infinitely many gaps in its graph. The result obtained by careful study spectral properties one-parametric family convolution operators L. Our results pure...

We give a necessary and sufficient condition for a locally compact group to be isomorphic to a closed cocompact subgroup in the isometry group of a Diestel–Leader graph. As a consequence of this condition, we see that every cocompact lattice in the isometry group of a Diestel–Leader graph admits a transitive, proper action on some other Diestel–Leader graph. We also give some examples of lattic...

We use elementary methods to compute the L-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk-Zuk [4]. The latter result was used by Grigorchuk-Linnell-Schick-Zuk [3] to pro...

We give a geometric proof based on recent work of Eskin, Fisher and Whyte that the lamplighter group Ln has infinitely many twisted conjugacy classes for any automorphism φ only when n is divisible by 2 or 3, originally proved by Gonçalves and Wong. We determine when the wreath product G o Z has this same property for several classes of finite groups G, including symmetric groups and some nilpo...

Abstract We prove the vanishing of bounded cohomology lamplighter groups for a wide range coefficients. This implies same number with self-similarity properties, such as Thompson’s group F . In particular, these are boundedly acyclic. Our method is ergodic and applies to “large” transformation where Mather–Matsumoto–Morita sometimes fails because not all acyclic in usual sense.