Characterization of Fuchsian groups containing hyperbolic elements I want to find the Fuchsian groups that acts on the upper half plane $\mathbb{H}$ to give $n$-holed torus $\mathbb{T_n}$. I am following the book Fuschian Groups by Svetlana Katok. There's this corollary-

Corollary 4.2.7. A Fuchsian group $\Gamma$ is cocompact if and only if $\mu(\mathbb{H}/\Gamma)<\infty$ and $\Gamma$ contains no parabolic elements.

where cocompact Fuschian groups are defined as follows-

Definition. A Fuchsian group is said to be cocompact if $\mathbb{H}/\Gamma$ is compact.

To get $\mathbb{T_n}$ as quotient $\Gamma$should act freely on $\mathbb{H}$. Therefore $\Gamma$ can't contain elliptic elements as elliptic elements fixes two points in $\mathbb{H}$. So by above corollary, the only choice we have is hyperbolic elements.
From here, I want to show that $\Gamma$ is genrated by $2n$ elements where $n\in \mathbb{N}−\{1\}$, more precisely $\Gamma = \langle a_1,b_1,...,a_n,b_n | a_1b_1a_1^{-1}b_1^{-1}...a_nb_na_n^{-1}b_n^{-1}=1\rangle $
 A: The book you are reading essentially contains an answer to your question. Start with your genus $n$ surface $S$ and represent it as  a 4n-gon $P$ with the standard identification. Recall also that $S=H^2/\Gamma$. The next step is a bit tricky: Lift the polygon $P$ to the hyperbolic plane, to a polygon $\tilde{P}$. There will be $2n$  elements $a_1, b_1...,a_{n}, b_n$ of $\Gamma$ which pair the sides of $\tilde{P}$. The existence of a lift  requires some work. For instance, if you know covering theory, use the fact that $P$ is simply-connected. Or use the monodromy principle from the complex analysis. It all depends on what kind of math you know. Now, apply Theorem 3.5.4 from Katok's book to conclude that $a_1, b_1...,a_{n}, b_n$ generate $\Gamma$. Katok proves this theorem assuming that $\tilde{P}$ is a Dirichlet domain of $\Gamma$, but it is not really needed for the proof. You only need the fact that $\tilde{P}$ is a fundamental domain. One can even compute the presentation of $\Gamma$ using $\tilde{P}$ and its side-pairing, but that would require reading a different book. I think, it is in Maskit's book "Kleinian Groups."  
