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

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