2011/09/27 by Gábor Elek, Elek, Gabor, Ádám Timár +1
Mathematics · #43A07 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1109.5863
openalex publication_date 2011/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be a countable group acting on a countable set X by permutations. We give a necessary and sufficient condition for the action to have a quasi-invariant mean with a given cocycle. This can be viewed as a combinatorial analogue of the condition for the existence of a quasi-invariant measure in the Borel case given by Miller. Then we show a geometric condition that guarantees that the corresponding action on the Stone-Čech compactification is Zimmer amenable. The geometric condition (weighted hyperfiniteness) resembles Property A. We do not know the exact relation between the two notions, however, we can show that amenable groups and groups of finite asymptotic dimension are weighted hyperfinite.