Amenability and paradoxical decompositions for pseudogroups and for discrete metric spaces

TL;DR

This paper explores amenability and paradoxical decompositions in groups, actions, and metric spaces, establishing equivalences among various conditions and introducing the concept of Tarski number for non-amenable group actions.

Contribution

It formalizes the connection between amenability, paradoxical decompositions, and F{46}lner conditions for pseudogroups and metric spaces, and introduces the minimal Tarski number for non-amenable actions.

Findings

Invariant mean existence is equivalent to the F{46}lner condition for pseudogroups.

For bounded perturbations of the identity, several conditions including isoperimetric and spectral are equivalent.

The minimal Tarski number for non-amenable group actions is at least 4, with some numerical estimates provided.

Abstract

This is an expostion of various aspects of amenability and paradoxical decompositions for groups, group actions and metric spaces. First, we review the formalism of pseudogroups, which is well adapted to stating the alternative of Tarski, according to which a pseudogroup without invariant mean gives rise to paradoxical decompositions, and to defining a F{\o}lner condition. Using a Hall-Rado Theorem on matchings in graphs, we show then for pseudogroups that existence of an invariant mean is equivalent to the F{\o}lner condition; in the case of the pseudogroup of bounded perturbations of the identity on a locally finite metric space, these conditions are moreover equivalent to the negation of the Gromov's so-called doubling condition, to isoperimetric conditions, to Kesten's spectral condition for related simple random walks, and to various other conditions. We define also the minimal Tarski number of paradoxical decompositions associated to a non-amenable group action (an integer $\ge 4$), and we indicate numerical estimates (Sections II.4 and IV.2). The final chapter explores for metric spaces the notion of supramenability, due for groups to Rosenblatt.