Distinct distances on hyperbolic surfaces

Abstract:

For any cofinite Fuchsian group $\Gamma \subset \mathrm {PSL}(2, \mathbb {R})$, 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 $\mathrm {PSL}(2, \mathbb {Z})$ with $\mu =[\mathrm {PSL}(2, \mathbb {Z}): \Gamma ]<\infty$, 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 $C>0$.