Amenable Actions of Nonamenable Groups

Since 1929 when von Neumann [vN29] introduced the notion of an invariant mean on a group (and more generally on a G-set) there is a permanent interest in the study of the phenomenon known as amenability. Amenable objects like groups, semigroups, algebras, graphs, metric spaces, operator algebras etc. play an important role in different areas of mathematics. A big progress in understanding of the structure of the class of amenable groups and in the study of asymptotic characteristics of them like growth of Følner sets (the notion introduced by A.M. Vershik in [Ver73]), drift, entropy etc. was reached in the past two decades [Ver73, Gri85, KV83, Gri98, CSGH99, BV, Ers04, Ers03, Ers05, BKNV04]. An important role in propaganda of the idea of amenability belongs to, perhaps the best, introductory to the subject of amenable groups book of Greenleaf [Gre69] where the following question is formulated.