  For any cofinite Fuchsian group $\Gamma\subset {\rm PSL}(2, \mathbb{R})$, we
show that any set of $N$ points on the hyperbolic surface
$\Gamma\backslash\mathbb{H}^2$ 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 $\Gamma\backslash\mathbb{H}^2$ determines $\geq C\frac{N}{\mu\log
N}$ distinct distances for some absolute constant $C>0$.
