Distinct distances on hyperbolic surfaces

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

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