On commutators and hyperbolic groups in {PSL(2, mathbb{R})}

A completion of Alan Beardon’s results on commutators and hyperbolic groups in ({PSL (2, mathbb{R}})) is given. It is proved geometrically that given ({g, h in PSL(2,mathbb{R})}) , two transformations that do not share fixed points, the commutator is always hyperbolic, unless a constant (depending on the translation lengths and the angle of intersection of the axes) is smaller than or equal to one (Theorem 3.3). This result allows to show that the inequality proved by Beardon $${sinh frac 12 rho (x, g(x)) sinh frac 1 2rho (x,h(x))geq 1,}$$ is indeed strict, where g, h generate a non elementary purely hyperbolic group (Theorem 4.2).