There are two kinds of classifications of Riemann surfaces.

Classification 1: Let $M$ be a Riemann surface. We will call $M$:

*elliptic*iff $M$ is compact (= closed);*parabolic*iff $M$ is not compact and $M$ doesn't carry a negative non-constant subharmonic function;*hyperbolic*iff $M$ does carry a negative non-constant subharmonic function.

Classification 2: Let $M$ be a Riemann surface. Let $\tilde{M}$ be the universal covering surface of $M$. We'll call $M$:

*elliptic*iff $\tilde{M}$ is conformally equivalent to $\mathbb{S}^2$;*parabolic*iff $\tilde{M}$ is conformally equivalent to $\mathbb{C}$;*hyperbolic*iff $\tilde{M}$ is conformally equivalent to the unit disk $\mathbb{D}$.

I want to know, what's the relationship between these two classifications? How to characterize an elliptic/parabolic/hyperbolic surface $M$ in the sense of the first classification by the corresponding Fuchsian group (I mean when $M=\mathbb{D}/\Gamma$ for some Fuchsian group $\Gamma$)?