Amenability and paradoxical decompositions for pseudogroups and for discrete metric spaces

By Tullio Ceccherini-Silberstein, Rostislav I. Grigorchuk, Pierre de la Harpe. 2016

132 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8eb931d22eeb9bac6f8b4186399270f15f841376

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Citations in the corpus

Combinatorial algebra : syntax and semantics (2014, unknown): “4), it is proved in [82] that the Tarski number of B(m,n), m,n as above, is between 6 and 14.”
Amenability via random walks (2003, background): “The classes of elementary amenable (EG), and subexponentially amenable (SG, see Grigorchuk (1998), and Ceccherini et al. (1999), §14) groups are the closure of (i), (ii) under ∗Department of Mathematics, University of California, Berkeley, CA 94720. laurent@math.berkeley.edu.”
Some topics in the dynamics of group actions on rooted trees (2011, background): “In the language of Cayley graphs associated with a group or of Schreier graphs associated with the action of a group, Følner’s condition means the existence of large subsets of the vertex set with a small boundary compared with the cardinality of the subset itself (here it does not matter how the bo”