Amenability, proximality and higher order syndeticity

TL;DR

This paper explores the structure of certain dynamical flows of discrete groups using higher order syndeticity, providing new characterizations and conditions for amenability and proximality.

Contribution

It introduces a higher order notion of syndeticity to characterize universal minimal proximal and strongly proximal flows of discrete groups.

Findings

Realization of universal flows as Stone spaces of specific Boolean algebras

New algebraic and topological characterizations of syndetic sets

Necessary and sufficient conditions for group amenability and strong amenability

Abstract

We show that the universal minimimal proximal flow and the universal minimal strongly proximal flow of a discrete group can be realized as the Stone spaces of translation invariant Boolean algebras of subsets of the group satisfying a higher order notion of syndeticity. We establish algebraic, combinatorial and topological dynamical characterizations of these subsets that we use to obtain new necessary and sufficient conditions for strong amenability and amenability. We also characterize dense orbit sets, answering a question of Glasner, Tsankov, Weiss and Zucker.