B. Hartley asked the following question: Does there exist a torsion free barely transitive group? Existence of torsion free simple barely transitive group is also unknown. We answer the latter question negatively in a special case. Moreover we proved the following: Let $G$ be a simple barely transitive group, and $H$ be a stabilizer of a point. If for a non-identity element $x \in G$, $C_G (x)$ is infinite then, $C_G (x)$ cannot contain $H$.
KUZUCUOĞLU, MAHMUT (2000) "On Torsion-Free Barely Transitive Groups," Turkish Journal of Mathematics: Vol. 24: No. 3, Article 7. Available at: https://journals.tubitak.gov.tr/math/vol24/iss3/7