Product set phenomena for countable groups

Countable Primitive Groups

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, oft...

Menger's Theorem for a Countable Source Set

Paul Erd} os has conjectured that Menger's theorem extends to innnite graphs in the following way: whenever A; B are two sets of vertices in an innnite graph, there exist a set of disjoint A{B paths and an A{B separator in this graph so that the separator consists of a choice of precisely one vertex from each of the paths. We prove this conjecture for graphs that contain a set of disjoint paths...

Rigidity theorems for actions of product groups and countable Borel equivalence relations

Plünnecke Inequalities for Countable Abelian Groups

We establish in this paper a new form of Plünnecke-type inequalities for ergodic probability measure-preserving actions of any countable abelian group. Using a correspondence principle for product sets, this allows us to deduce lower bounds on the upper and lower Banach densities of any product set in terms of the upper Banach density of an iterated product set of one of its addends. These boun...

Mf-property for Countable Discrete Groups

In this article we study MF-property for countable discrete groups, i.e. groups which admit embedding into unitary group of C∗-algebra ∏ Mn/ ⊕ Mn. We prove that Baumslag group 〈a, b|aab = a2〉 has MF-property and check some permanent facts about MF-groups.