Distinct distances on hyperbolic surfaces

Abstract

For any cofinite Fuchsian group Γ ⊂ PSL(2, R), we show that any set of N points on the hyperbolic surface Γ\H 2 determines ≥ C Γ 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 PSL(2, Z) with µ = [PSL(2, Z) : Γ] < ∞, any set of N points on Γ\H 2 determines ≥ C N µ log N distinct distances for some absolute constant C > 0.