On the minimal distance between elliptic fixed points for geometrically-finite Fuchsian groups

Let $\mathrm{\Gamma }$ be a geometrically-finite Fuchsian group acting on the upper half plane $\text{hh}\mathrm{.}$ Let $\text{E}$ denote the set of elliptic fixed points of $\mathrm{\Gamma }$ in $\text{hh}\mathrm{.}$ We give a lower bound on the minimal hyperbolic distance between points in $\text{E}\mathrm{.}$ Our bound depends on a universal constant and the length of the smallest closed geodesic on $\mathrm{\Gamma }\mathrm{\backslash }\text{hh}\mathrm{.}$