How to Prove a Riemann Surface is Hyperbolic by Showing its Volume Grows Exponentially?

Question

We have a Holomorphic Foliation by curves on a Complex Manifold. Therefore, every leaf is a Riemann Surface.

Let $R$ be a Riemann Surface.

We say that the Riemann Surface $R$ is Hyperbolic if and only if one of the two following equivalent conditions is satisfied:

1. $R$ is isomorphic to a quotient of the Upper Half-Plane by a Fuchsian Group.
2. Its covering space is isomorphic to the Hyperbolic Disk.

Meanwhile, it is mentioned that to prove that the Riemann Surface $R$ is Hyperbolic it is sufficient to prove its volume grows exponentially.

What does that mean?

(Or, How to define the Volume of the Riemann Surface $R$, and to find out whether or not the volume grows exponentially?).

Answer 1

I'll use the word "area" instead of "volume", since we are in 2 dimensions.

The Uniformization Theorem implies that there exists a complete Riemannian metric on $R$ in the given conformal class with constant curvature $\kappa = -1$, $0$ or $+1$. Furthermore, that metric is unique if $\kappa = -1$ or $+1$, and it is unique up to scaling if $\kappa = 0$. Area on $R$ is defined using the area form of that metric.

In the case $\kappa = 1$, $R$ is a sphere and has finite area.

In the case $\kappa = 0$, $R$ is either a torus which has finite area, a cylinder which has linearly growing area, or the Euclidean plane which has quadratically growing area.

It follows that if the area of $R$ grows exponentially then $\kappa = -1$.