Is there a highly transitive action of a finitely generated torsion simple group? 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 $1 \leq i \leq k$.
 A: Yes, such groups exist. Simple groups in my paper "Palindromic subshifts and simple periodic groups of intermediate growth" are like that.
A: As you expressed interest in Thompson's group V in the comments --
the following is a highly transitive faithful permutation representation of V on $\mathbb{Z}$ (although as Ives de Cornulier has already mentioned this group is not
a torsion group):
Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$ which interchanges
$r_1+km_1$ and $r_2+km_2$ for every $k \in \mathbb{Z}$ and which fixes everything else.
Then we have $V \cong \langle \kappa, \lambda, \mu, \nu \rangle$, where
$\kappa := \tau_{0(2),1(2)}$, $\lambda := \tau_{1(2),2(4)}$, $\mu := \tau_{0(2),1(4)}$
and $\nu := \tau_{1(4),2(4)}$ satisfy the defining relations


*

*$\kappa^2 = \lambda^2 = \mu^2 = \nu^2 = 1$,

*$\lambda\kappa\mu\kappa\lambda\nu\kappa\nu\mu\kappa\lambda\kappa\mu = 1$,

*$\kappa\nu\lambda\kappa\mu\nu\kappa\lambda\nu\mu\nu\lambda\nu\mu = 1$,

*$(\lambda\kappa\mu\kappa\lambda\nu)^3 = (\mu\kappa\lambda\kappa\mu\nu)^3 = 1$,

*$(\lambda\nu\mu)^2\kappa(\mu\nu\lambda)^2\kappa = 1$,

*$(\lambda\nu\mu\nu)^5 = 1$,

*$(\lambda\kappa\nu\kappa\lambda\nu)^3\kappa\nu\kappa(\mu\kappa\nu\kappa\mu\nu)^3 \kappa\nu\kappa\nu = 1$,

*$((\lambda\kappa\mu\nu)^2(\mu\kappa\lambda\nu)^2)^3 = 1$,

*$(\lambda\nu\lambda\kappa\mu\kappa\mu\nu\lambda\nu\mu\kappa\mu\kappa)^4 = 1$,

*$(\mu\nu\mu\kappa\lambda\kappa\lambda\nu\mu\nu\lambda\kappa\lambda\kappa)^4 = 1$,

*$(\lambda\mu\kappa\lambda\kappa\mu\lambda\kappa\nu\kappa)^2 = 1$, and

*$(\mu\lambda\kappa\mu\kappa\lambda\mu\kappa\nu\kappa)^2 = 1$
given in:
Graham Higman, Finitely presented infinite simple groups, Notes on Pure
Mathematics, Department of Pure Mathematics, Australian National University,
Canberra, 1974. MR0376874.
