Geodesic cover of Fuchsian groups

TL;DR

This paper investigates unions of fundamental domains, called geodesic covers, of Fuchsian groups, revealing that their finiteness characterizes geometrically finite hyperbolic surfaces, with proofs based on Shimizu's lemma.

Contribution

It introduces the concept of geodesic covers for Fuchsian groups and establishes their finiteness as a characterization of geometrical finiteness.

Findings

Finiteness of geodesic covers characterizes geometrically finite Fuchsian groups.

Uses Shimizu's lemma to prove results in the geometrically finite case.

Provides a new perspective on the structure of hyperbolic surfaces.

Abstract

We study unions of fundamental domains of a Fuchsian group, especially those with hyperbolic plane metric realizing the metric of the corresponding hyperbolic surface. We call these unions the \textit{geodesic covers} of the Fuchsian group or the hyperbolic surface. The paper contributes to showing that finiteness of geodesic covers is basically another characterization of geometrically finiteness. The resolution of geometrically finite case is based on Shimizu's lemma.