Amenable, transitive and faithful actions of groups acting on trees

TL;DR

This paper investigates conditions under which groups acting on trees, such as amalgamated free products and HNN-extensions, admit amenable, transitive, and faithful actions on countable sets, extending known results using Baire category arguments.

Contribution

It establishes new criteria for groups acting on trees to have amenable, transitive, and faithful actions, broadening understanding of group actions in this context.

Findings

Existence of such actions if initial groups have amenable, almost free actions with infinite orbits

Extension of results to groups acting on trees

Use of Baire category Theorem in the analysis

Abstract

We study under which condition an amalgamated free product or an HNN-extension over a finite subgroup admits an amenable, transitive and faithful action on an infinite countable set. We show that such an action exists if the initial groups admit an amenable and almost free action with infinite orbits (e.g. virtually free groups or infinite amenable groups). Our result relies on the Baire category Theorem. We extend the result to groups acting on trees.