Amenable actions of amalgamated free products of free groups over a cyclic subgroup and generic property

TL;DR

This paper demonstrates that amalgamated free products of free groups over a cyclic subgroup can act amenably, faithfully, and transitively on infinite countable sets, extending previous results on doubles of free groups.

Contribution

It generalizes known results by establishing amenable actions for a broader class of amalgamated free products of free groups.

Findings

Existence of amenable, faithful, and transitive actions on infinite sets

Extension of previous results from free group doubles to more general amalgamated products

Broader understanding of group actions in geometric group theory

Abstract

We show that the class of amalgamated free products of two free groups over a cyclic subgroup admits amenable, faithful and transitive actions on infinite countable sets. This work generalizes the results on such actions for doubles of free group on 2 generators.