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:


1. Why $\pi\_1(S)\cong \mathbb{Z}\ $ implies the volume of $S\ $ is infinite?
2. Is there a general relation between the fundamental group and the volume of a surface?


Can someone help me, please? 



Volume of a Riemannian manifold and its relation to fundamental group
Hyperbolic surfaces are of the form $\mathbb{H}^2/\Gamma$, where $\Gamma$ is a Fuchsian group which is isomorphic to the fundamental group. If $\Gamma \cong \mathbb{Z}$, then it is generated by a single element $\gamma$. The order of $\gamma$ is infinite, so it cannot be elliptic. It is thus either parabolic (and hence conjugate to 
$\begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}$ for some $\lambda$) or hyperbolic (and hence conjugate to $\begin{pmatrix} \lambda & 0 \\ 0 & 1/\lambda \end{pmatrix}$ for some $\lambda \neq 0$). It is an easy exercise to show that the quotient of $\mathbb{H}^2$ by the subgroup generated by $\begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}$ has infinite volume, and similarly for $\begin{pmatrix} \lambda & 0 \\ 0 & 1/\lambda \end{pmatrix}$.


For compact surfaces with Riemannian metrics, the Gauss-Bonnet theorem determines the volume in terms of the Euler characteristic (and hence in terms of the fundamental group).


I recommend reading Katok's book "Fuchsian Groups" if you want to acquire comfort with the above types of arguments.


