Distinct distances on surfaces arising from cofinite Fuchsian groups
Abstract
Erdős (1946) proposed the question of finding the minimal number of distinct distances among any N points in the plane. We consider this problem in hyperbolic surfaces associated with cofinite Fuchsian groups, i.e. the volume of the surface is finite. We prove a lower bound of the same strength as Guth-Katz. In particular, for any finite index subgroup of the modular group, we extract out the dependence of the implied constant on the index.