Liouville property of strongly transitive actions

TL;DR

This paper explores the Liouville property in strongly transitive group actions, introduces the n-Liouville concept, and proves it for small n, providing new insights into non-amenability and additive combinatorics.

Contribution

It defines the n-Liouville property for group actions, reformulates it using additive combinatorics, and proves it for n=1, 2, advancing understanding of group amenability.

Findings

Proved n-Liouville property for n=1, 2

Reformulated n-Liouville property in additive combinatorics

Identified open problem for n≥3

Abstract

Liouville property of actions of discrete groups can be reformulated in terms of existence co-F$\o$lner 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 $n\geq 3$ remains open.