The theorem of Dekking and Host [6] regarding tightness around the mean of first passage percolation on the binary tree, from the root to a boundary of a ball, is generalized to a class of graphs which includes all lattices in hyperbolic spaces and the lamplighter graph over N. This class of graphs is closed under product with any bounded degree graph. Few open problems and conjectures are gath...

In this paper, we prove that certain spaces are not quasi-isometric to Cayley graphs of finitely generated groups. In particular, we answer a question of Woess and prove a conjecture of Diestel and Leader by showing that certain homogeneous graphs are not quasi-isometric to a Cayley graph of a finitely generated group. This paper is the first in a sequence of papers proving results announced in...

An important conjecture in percolation theory is that almost surely no infinite cluster exists in critical percolation on any transitive graph for which the critical probability is less than 1. Earlier work has established this for the amenable cases Z and Z for large d, as well as for all non-amenable graphs with unimodular automorphism groups. We show that the conjecture holds for several cla...

‎we investigate two constructions‎ - ‎the replacement and the zig-zag‎ ‎product of graphs‎ - ‎describing several fascinating connections‎ ‎with combinatorics‎, ‎via the notion of expander graph‎, ‎group‎ ‎theory‎, ‎via the notion of semidirect product and cayley graph‎, ‎and‎ ‎with markov chains‎, ‎via the lamplighter random walk‎. ‎many examples‎ ‎are provided‎.

It is shown that certain ascending HNN extensions of free abelian groups of finite rank, as well as various lamplighter groups, can be realized as automaton groups, i.e., can be given a self-similar structure. This includes the solvable Baumslag-Solitar groups BS(1, m), for m 6= ±1. In addition, it is shown that, for any relatively prime integers m, n ≥ 2, the pair of Baumslag-Solitar groups BS...

We exhibit a family of infinite, finitely-presented, nilpotent-byabelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic Lie group. In addition, we propose a candidate for a polycyclic group whose q...

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