Erd\H{o}s distinct distances in hyperbolic surfaces

Abstract

In this paper, we introduce the notion of "geodesic cover" for Fuchsian groups, which summons copies of fundamental polygons in the hyperbolic plane to cover pairs of representatives realizing distances in the corresponding hyperbolic surface. Then we use estimates of geodesic-covering numbers to study the distinct distances problem in hyperbolic surfaces. Especially, for $Y$ from a large class of hyperbolic surfaces, we establish the nearly optimal bound $\geq c(Y)N/\log N$ for distinct distances determined by any $N$ points in $Y$, where $c(Y)>0$ is some constant depending only on $Y$. In particular, for $Y$ being modular surface or standard regular of genus $g\geq 2$, we evaluate $c(Y)$ explicitly. We also derive new sum-product type estimates.