A measure-scaling quasi-isometry between two connected graphs is a that quasi- $$\kappa $$ -to-one in natural sense for some >0$$ . For non-amenable graphs, all quasi-isometries are any , while amenable ones there exists at most one possible such an graph X, we show the set of forms subgroup $$\mathbb {R}_{>0}$$ call (measure-)scaling group X. This invariant under quasi-isometries. In context C...

We show that the measure on markings of Zn, d ≥ 3, with elements of {0,1} given by i.i.d. fair coin flips on the range R of a random walk X run until time T and 0 otherwise becomes indistinguishable from the uniform measure on such markings at the threshold T = 1 2 Tcov(Z d n). As a consequence of our methods, we show that the total variation mixing time of the random walk on the lamplighter gr...

Let L X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the p-norms and spectral radii of P , if one has an amenable (not necessarily discrete or unimodular) locally compact group of isome...

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. 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 simple random walk, as shown by Virág (2000). We ...

We study internal Lie algebras in the category of subshifts on a fixed group – or algebraic for short. We show that if acting is virtually polycyclic and underlying vector space has dense homoclinic points, such can be recoded to have cellwise bracket. On other hand there exist (on any finitely-generated non-torsion group) with operations whose bracket cannot cellwi...