# Triple transitivity and non-free actions in dimension one

## TL;DR

This paper classifies all 3-transitive actions of certain groups acting on the circle or a tree, providing new insights into the transitivity degrees of these groups and their actions.

## Contribution

It introduces a classification of 3-transitive actions for groups acting on the circle or a tree, linking dynamics to transitivity degree constraints.

## Key findings

- All faithful 3-transitive actions are conjugate to actions on orbits in the circle or boundary of the tree.
- Provides new classes of groups with computable transitivity degrees.
- Establishes conditions under which higher transitivity actions are essentially unique.

## Abstract

The transitivity degree of a group $G$ is the supremum of all integers $k$ such that $G$ admits a faithful $k$-transitive action. Few obstructions are known to impose an upper bound on the transitivity degree for infinite groups. The results of this article provide two new classes of groups whose transitivity degree can be computed, as a corollary of a classification of all $3$-transitive actions of these groups. More precisely, suppose that $G$ is a subgroup of the homeomorphism group of the circle $\mathsf{Homeo}(\mathbb{S}^1)$ or the automorphism group of a tree $\mathsf{Aut}(\mathbb{T})$. Under natural assumptions on the stabilizers of the action of $G$ on $\mathbb{S}^1$ or $\partial \mathbb{T}$, we use the dynamics of this action to show that every faithful action of $G$ on a set that is at least $3$-transitive must be conjugate to the action of $G$ on one of its orbits in $\mathbb{S}^1$ or $\partial \mathbb{T}$.