Consider the upper half plane $\mathbb{H}$ and a Fuchsian group $\Gamma$ (i. e. a discrete subgroup of $PSL(2,\mathbb{R})$ that acts freely and properly discontinuously on $\mathbb{H}$.
I want to show that if $\mathbb{H}/\Gamma$ is compact then $\Gamma$ is genrated by $2n$ number of elements where $n\in \mathbb{N}-\{1\}$.
My attempt :
$PSL(2,\mathbb{R})$ is isomorphic to the group of all Möbius transformations $f$ on $\mathbb{H}$ where $f(z)=\frac{az+b}{cz+d}; a,b,c,d\in \mathbb{R}, ad-bc=1$. Since elliptic elements (i.e. elements with $(a+d)^2<4$) fix $2$ points of $\mathbb{H}$, $\Gamma$ does not contain elliptic elements.
Also a corollary from the book Fuchsian Groups by Svetlana Katok says-
Corollary 4.2.7. A Fuchsian group $G$ is cocompact if and only if $\mu(\mathbb{H}/G)<\infty$ and $G$ contains no non-identity parabolic elements.
where a Fuchsian group $G$ is said to be cocompact if $\mathbb{H}/G$ is compact.
Therefore $\Gamma$ does not contain non-identity parabolic elements (i.e. elements with $(a+b)^2=4$).
Thus, so far what I could get is $\Gamma$ can contain only hyperbolic elements (i. e. elements with $(a+d)^2>4$). I have no idea how to proceed further. How can one prove that $\Gamma$ is generated by $2n$ elements where $n\in \mathbb{N}-\{1\}$.