Amenability and paradoxical decompositions for pseudogroups and for   discrete metric spaces
  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. 
Tullio Ceccherini-Silberstein, Rostislav I. Grigorchuk, and Pierre de   la Harpe
math.GR math.DS math.FA math.MG