On the displacement of generators of free Fuchsian groups

We prove an inequality that must be satisfied by displacement of generators of free Fuchsian groups, which is the two-dimensional version of the log(2k-1) Theorem for Kleinian groups due to Anderson et al. (J Differ Geom 44:738-782, 1996). As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as the two-dimensional Margulis constant and lengths of a pair of based loops, which improves a result of Buser's.