Permanent properties of amenable, transitive and faithful actions

TL;DR

This paper investigates the hereditary properties of countable groups that admit amenable, transitive, and faithful actions, focusing on amalgamated free products and showing specific constructions that possess these actions.

Contribution

It demonstrates that doubles of amenable groups and amalgamated free products over finite subgroups admit amenable, transitive, and faithful actions, expanding understanding of these properties.

Findings

Double of an amenable group admits such an action.

Amalgamated free products over finite subgroups admit such actions.

Hereditary properties are preserved in these constructions.

Abstract

We study hereditary properties of the class of countable groups admitting an amenable, transitive and faithful action on a countable set. We consider mainly the case of amalgamated free products, and we show in particular that the double of amenable groups and the amalgamated free products of two amenable groups over a finite subgroup admit such actions.