2018/06/07 by Juschenko, Kate
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1806.02753
Liouville property of actions of discrete groups can be reformulated in terms of existence co-Følner sets. Since every action of amenable group is Liouville, the property can be served as an approach for proving non-amenability. The verification of this property is conceptually different than finding a non-amenable action. 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-Liouville property of action to be Liouville property of point-wise action of the group on the sets of cardinality n. We reformulate n-Liouville property in terms of additive combinatorics and prove it for n=1, 2. The case n≥ 3 remains open.