I am reading a book (Mapping Class Group by Farb and Margalit) and it says (in a proof of one theorem):
If $S$ admits a hyperbolic metric (they define such a surface to be of finite area and complete) and we had $\pi_1(S)\cong \mathbb{Z}\ $ then the surface would have an infinite volume which is a contradiciton. Hence $\pi_1(S)$ is NOT isomorphic to $\mathbb{Z}$.
Questions
- If we have a Riemannian manifold of finite area, does it have a finite volume also? I am interested in the case of hyperbolic surfaces.
- Why $\pi_1(S)\cong \mathbb{Z}\ $ implies the volume of $S\ $ is infinite?
Can someone help me, please?