An $n$-holed torus $\mathbb{T_n}$ is quotient of the upper half plane $\mathbb{H}$ by properly discontinuous action of a subgroup of automorphism group of $\mathbb{H}$. The automorphism group of $\mathbb{H}$ is precisely $PSL(2,\mathbb{R})$. Also, a subgroup of $PSL(2,\mathbb{R})$ acts properly discontinuously if and only if it is discrete i.e. a Fuchsian group. Thus now we have $\mathbb{T_n}$ is quotient of $\mathbb{H}$ by action of a Fuchsian group, say, $\Gamma$. Moreover, the fundamental group $\pi_1(\mathbb{H}/ \Gamma)$ is isomorphic to $\Gamma$. Therefore,
$\\ \pi_1(\mathbb{H}/ \Gamma)=$ $\pi_1(\mathbb{T_n})=$ $\langle a_1,b_1,...,a_n,b_n | a_1b_1a_1^{-1}b_1^{-1}...a_nb_na_n^{-1}b_n^{-1}=1 \rangle$
So the Fuchsian group, $\Gamma=\langle a_1,b_1,...,a_n,b_n | a_1b_1a_1^{-1}b_1^{-1}...a_nb_na_n^{-1}b_n^{-1}=1 \rangle$
I am unable to see how does $\Gamma $ exactly act on $\mathbb{H}$. How can one give the action in terms of elements of $\Gamma $ and $\mathbb{H}$?