  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.
