It is proved that the associated Lie algebra of Mal’tsev ?-completion lamplighter group pronilpotent completion algebra. also shown homology this completed uncountable dimension in each degree.

We construct a bireversible self-dual automaton with 3 states over an alphabet with 3 letters which generates the lamplighter group Z3 ≀ Z. 2010 Mathematics Subject Classification: 20F65, 20M35

In this note we explain how the computation of the spectrum of the lamplighter group from [GZ01] yields a counterexample to a strong version of the Atiyah conjectures about the range of L -Betti numbers of closed manifolds.

First, we answer a question of Pestov, by proving that the full group of a sofic equivalence relation is a sofic group. Then, we give a short proof of the theorem of Grigorchuk and Medynets that the topological full group of a minimal Cantor homeomorphism is LEF. Finally, we show that for certain non-amenable groups all the generalized lamplighter groups are sofic.

We consider a programming language based on the lamplighter group that uses only composition and iteration as control structures. We derive generating functions and counting formulas for this language with the goal of avoiding repetition of equivalent programs.

In this paper we introduce a way to estimate a level of closeness of Cayley automatic groups to the class of automatic groups using a certain numerical characteristic. We characterize Cayley automatic groups which are not automatic in terms of this numerical characteristic and then study it for the lamplighter group, the Baumslag–Solitar groups and the Heisenberg group.

We develop the geometry of the Cayley graph of the lamplighter group with respect to the generating set rising from its interpretation as an automata group [4]. We find metric behavior with respect to this generating set analogous to the metric behavior in the standard group theoretic generating set. This includes expressions for normal forms and geodesic paths, and families of ‘dead-end’ words...

Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as “lamplighter groups”. We prove that zero speed for random walk on a lamplig...

We show that the type function of a space with finite asymptotic dimension estimates its Hilbert (or any l) compression. The method allows to obtain the lower bound of the compression of the lamplighter group Z≀Z, which has infinite asymptotic dimension.