Journal of the
Korean Mathematical Society JKMS

In this article, we will characterize structures of geometric quotient orbifolds of $G$-manifold of genus two where $G$ is a finite group of orientation preserving diffeomorphisms using the idea of handlebody orbifolds. By using the characterization, we will deduce the candidates of possible non-hyperbolic geometric quotient orbifolds case by case using W. Dunbar's work. In addition, if the $G$-manifold is compact, closed and the quotient orbifold's geometry is hyperbolic then we can show that the fundamental group of the quotient orbifold cannot be in the class $\mathcal{D}$.

Keywords: orbifold, finite group action, handlebody orbifold, Heegaard splitting

MSC numbers: Primary 57M99; Secondary 57S17