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:

Why $\pi_1(S)\cong \mathbb{Z}\ $ implies the volume of $S\ $ is infinite?

Is there a general relation between the fundamental group and the volume of a surface?

Can someone help me, please?