We explore the geometry of the Cayley graphs of the lamplighter groups and a wide range of wreath products. We show that these groups have dead end elements of arbitrary depth with respect to their natural generating sets. An element w in a group G with finite generating set X is a dead end element if no geodesic ray from the identity to w in the Cayley graph Γ(G, X) can be extended past w. Add...

Given a finite graph G, a vertex of the lamplighter graph G♦ = Z2 o G consists of a zero-one labeling of the vertices of G, and a marked vertex of G. For transitive G we show that, up to constants, the relaxation time for simple random walk in G♦ is the maximal hitting time for simple random walk in G, while the mixing time in total variation on G♦ is the expected cover time on G. The mixing ti...

Benjamini, Lyons and Schramm [Random Walks and Discrete Potential Theory (1999) 56–84] considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. In this paper we solve several problems raised by those authors. The anchored expansion constant is a variant of the Cheeger constant; its positivity implies positive lower speed for the...

Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups Gn that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his con...

Let (G, μ) be a discrete group with a generating probability measure. Nevo shows that if G has property (T) then there exists an ε > 0 such that the Furstenberg entropy of any (G, μ)-stationary space is either zero or larger than ε. Virtually free groups, such as SL2(Z), do not have property (T). For these groups, we construct stationary actions with arbitrarily small, positive entropy. This co...

The lamplighter group over Z is the wreath product Zq ≀ Z. With respect to a natural generating set, its Cayley graph is the Diestel-Leader graph DL(q, q). We study harmonic functions for the “simple” Laplacian on this graph, and more generally, for a class of random walks on DL(q, r), where q, r ≥ 2. The DL-graphs are horocyclic products of two trees, and we give a full description of all posi...

We consider Turing machines as actions over configurations in ΣZ d which only change them locally around a marked position that can move and carry a particular state. In this setting we study the monoid of Turing machines and the group of reversible Turing machines. We also study two natural subgroups, namely the group of finite-state automata, which generalizes the topological full groups stud...

Abstract In this work, we study automorphisms of synchronous self-similar groups and the existence extensions to continuous over closure these with respect depth metric. We obtain conditions for continuity such extensions, but also construct examples where do not exist. in detail case lamplighter group <m:mi mathvariant="sc...