Distinct distances on hyperbolic surfaces

For any cofinite Fuchsian group Γ ⊂ mathvariant="normal">P mathvariant="normal">S mathvariant="normal">L ( 2 , mathvariant="double-struck">R stretchy="false">) \Gamma \subset \mathrm {PSL}(2, \mathbb {R}) , we show that set of alttext="upper N"> N encoding="application/x-tex">N points on the hyperbolic surface minus H squared"> class="MJX-variant" mathvariant="normal">∖<!-- ∖ <mml:msup> mathvariant="double-struck">H \backslash {H}^2 determines alttext="greater-than-or-equal-to C Subscript Baseline StartFraction N Over log EndFraction"> ≥<!-- ≥ <mml:msub> C log ⁡<!-- ⁡ </mml:mfrac> encoding="application/x-tex">\geq C_{\Gamma } \frac {N}{\log N} distinct distances for some constant greater-than 0"> &gt; 0 encoding="application/x-tex">C_{\Gamma }&gt;0 depending only Gamma"> encoding="application/x-tex">\Gamma . In particular, being finite index subgroup Z mathvariant="double-struck">Z encoding="application/x-tex">\mathrm {Z}) with alttext="mu equals left-bracket right-parenthesis colon right-bracket infinity"> μ<!-- μ <mml:mo>= stretchy="false">[ : stretchy="false">] mathvariant="normal">∞<!-- ∞ encoding="application/x-tex">\mu =[\mathrm {Z}): \Gamma ]&gt;\infty mu C\frac {N}{\mu \log absolute encoding="application/x-tex">C&gt;0