Does there exist a fuchsian group that is not hyperbolic ?
By hyperbolic group I mean a group whose Cayley graph is hyperbolic in the sense of Gromov, and a fuchsian group is a discrete subgroup of PSL$_2(\mathbb{R})$.
The fuchsian groups that are cocompact are definitely hyperbolic, but maybe there exists a non cocompact fuchsian group that is not hyperbolic... maybe it could even be finitely generated.