Uniform amenability

Abstract: According to the classical result of Connes, Feldman and Weiss, measured hyperfiniteness of a group action is equivalent to measured amenability. In the Borel category it is known that hyperfiniteness implies amenability and it is conjectured that the converse is true. Based on the work of Anantharaman-Delaroche and Renault, one can introduce the notion of uniform amenability, a strengthening of measured amenability (it is a sort of exactness in the category of measurable actions, so the famous Gromov-Osajda groups have no free uniformly amenable actions). One can also introduce the notion of uniform hyperfiniteness in a rather natural way. We prove that the two notions are equivalent provided that the measurable action satisfies a boundedness condition for the Radon-Nikodym derivative (e.g. in the case of Poisson boundaries).

combinatorics

Audience: researchers in the topic

Warwick Combinatorics Seminar

Series comments: This is the online combinatorics seminar at Warwick.