We prove that a lamplighter graph of a locally finite graph over a finite graph does not admit a non-constant harmonic function of finite Dirichlet energy.

In this note we give three short results concerning the elliptic Harnack inequality (EHI), in the context of random walks on graphs. The first is that the EHI implies polynomial growth of the number of points in balls, and the second that the EHI is equivalent to an annulus type Harnack inequality for Green’s functions. The third result uses the lamplighter group to give a counterexample concer...

We define the notion of uniformly recurrent subgroup, URS in short, which is a topological analog of the notion of invariant random subgroup (IRS), introduced in [2]. Our main results are as follows. (i) It was shown in [26] that for an arbitrary countable infinite group G, any free ergodic probability measure preserving G-system admits a minimal model. In contrast we show here, using URS’s, th...

We calculate the spectra and spectral measures associated to random walks on restricted wreath products G wr Z, with G a finite abelian group, by realizing them as groups generated by automata. This generalizes the work of Grigorchuk and Żuk on the lamplighter group. More generally we calculate the spectra of random walks on groups generated by Cayley machines of finite groups and calculate Kes...

We build presentations for automata groups generated by Cayley machines of finite nilpotency class two and prove that these are all cross-wired lamplighter groups.

Let T1, . . . , Td be homogeneous trees with degrees q1 + 1, . . . , qd + 1 ≥ 3, respectively. For each tree, let h : Tj → Z be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of T1, . . . , Td is the graphDL(q1, . . . , qd) consisting of all d-tuples x1 · · ·xd ∈ T1×· · ·×Td with h(x1) + · · ·+ h(xd) = 0, equipped wi...

We determine all positive harmonic functions for a large class of “semiisotropic” random walks on the lamplighter group, i.e., the wreath product Zq ≀Z, where q ≥ 2. This is possible via the geometric realization of a Cayley graph of that group as the Diestel-Leader graph DL(q, q). More generally, DL(q, r) (q, r ≥ 2) is the horocyclic product of two homogeneous trees with respective degrees q+1...

We compute the Bieri-Neumann-Strebel-Renz geometric invariants, Σn, of the lamplighter groups Lm by using the Diestel-Leader graph DL(m,m) to represent the Cayley graph of Lm.

We compute the von Neumann dimensions of the kernels of adjacency operators on free lamplighter groups and show that they are irrational, thus providing an elementary constructive answer to a question of Atiyah.