Non-abelian free groups admit non-essentially free actions on rooted trees

TL;DR

This paper demonstrates that non-abelian free groups can act on rooted trees in a way that is transitive but not essentially free on the boundary, answering a longstanding question in group actions.

Contribution

It constructs explicit actions of non-abelian free groups on rooted trees with non-essentially free boundary actions, extending previous results and resolving an open question.

Findings

Non-abelian free groups admit non-essentially free actions on rooted trees.

Such actions are spherically transitive on the tree.

The boundary action of these groups is not essentially free.

Abstract

We show that every countable non-abelian free group $\Gamma $ admits a spherically transitive action on a rooted tree $T$ such that the action of $\Gamma $ on the boundary of $T$ is not essentially free. This reproves a result of Bergeron and Gaboriau. The existence of such an action answers a question of Grigorchuk, Nekrashevich and Sushchanskii.