Distinct distances on hyperbolic surfaces

For any cofinite Fuchsian group $Gammasubset {rm PSL}(2, mathbb{R})$, we show that any set of $N$ points on the hyperbolic surface $Gammabackslashmathbb{H}^2$ determines $geq C_{Gamma} frac{N}{log N}$ distinct distances for some constant $C_{Gamma}>0$ depending only on $Gamma$. In particular, for $Gamma$ being any finite index subgroup of ${rm PSL}(2, mathbb{Z})$ with $mu=[{rm PSL}(2, mathbb{Z}): Gamma ]<infty$, any set of $N$ points on $Gammabackslashmathbb{H}^2$ determines $geq Cfrac{N}{mulog N}$ distinct distances for some absolute constant $C>0$.