Distinct distances on hyperbolic surfaces

TL;DR

This paper establishes a lower bound on the number of distinct distances determined by N points on hyperbolic surfaces associated with cofinite Fuchsian groups, extending classical distance problems to hyperbolic geometry.

Contribution

It proves a lower bound of order N / log N for the number of distinct distances on hyperbolic surfaces for groups with finite index in PSL(2, Z), generalizing Euclidean distance results.

Findings

At least C_Γ * N / log N distinct distances for cofinite Fuchsian groups

Special case for subgroups of PSL(2, Z) with bounds involving the index μ

Results extend distance problems to hyperbolic geometry contexts

Abstract

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