Liouville property of strongly transitive actions

Abstract

Liouville property of actions of discrete groups can be reformulated in terms of existence of co-Følner sets. Since every action of amenable group is Liouville, the property can be used for proving non-amenability. There are many groups that are defined by strongly transitive actions. In some cases amenability of such groups is an open problem. We define n n -Liouville property of action to be Liouville property of point-wise action of the group on the sets of cardinality n n . We reformulate n n -Liouville property in terms of additive combinatorics and prove it for n = 1 , 2 n=1, 2 . The case n ≥ 3 n\geq 3 remains open.