***Is there a highly transitive action of a finitely generated torsion simple group $G$ on $\mathbb{Z}$ ?***


Highly transitive means $k$-transitive for each $k \in \mathbb{N}$, that is: for every two $k$-tuples $(x\_1, \dots x\_k) , (y\_1, \dots, y\_k) \in \mathbb{Z}^k$ of pairwise distinct elements, there is some $g \in G$ such that $g(x\_i) = y\_i$ for each $1 \leq i \leq k$.



Is there a highly transitive action of a finitely generated torsion simple group?
Yes, such groups exist. Simple groups in my paper "Palindromic subshifts and simple periodic groups of intermediate growth" are like that.


