Triple transitivity and non‐free actions in dimension one

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