$S_n$ and the left half-plane $H^-$ covers the annular region $A^-$ on $S_n$.

Given $|\alpha_0|=a$, let $\Gamma_0$ be the group generated by the Möbius transformation $g_0(z)=e^az.$ Observe that the imaginary axis, denoted by $I$, projects to the geodesic $\alpha_0$ on $A={\Bbb{H}}^2/\Gamma_0$ and the left and right half-planes project, respectively, to the subannuli $A^-$ and $A^+$.

Define the sequence of Fuchsian groups recursively. Suppose the groups $\Gamma_0\subset\ldots\subset\Gamma_n$ have been defined. Set $E_{n+1}^+= E_{n+1}\cap A^+.$ Remove the subsurface $E_{n+1}^+$ from $S_n$ to get the surface $S'_n.$ A loop on $S'_n$ is homotopically trivial on $S'_n$ if and only if it is homotopically trivial on $S_n.$ Therefore, the preimage $\tilde{S}'_n$ of $S'_n$ under the covering projection $\pi_n:{\Bbb{H}}^2\rightarrow S_n$ is a connected and simply connected, $\Gamma_n$-invariant set and the restriction $\pi_n:\tilde{S}'_n\rightarrow S'_n$ is the universal covering.

Suppose $G_{n+1}$ is a Fuchsian group representing $S_{n+1}$. Let $F_{n+1}^+= F^*_{n+1}\cap A^+.$ Then remove $F_{n+1}^+$ from $S_{n+1}$ to get the surface $S^+_{n+1}.$ Let $\tilde{S}^+_{n+1}$ be a connected preimage of $S^+_{n+1} $ under the covering projection $\pi_{n+1}:{\Bbb{H}}^2\rightarrow S_{n+1}.$ Then $\tilde{S}^+_{n+1}$ is simply connected and the restriction $\pi_{n+1}:\tilde{S}^+_{n+1}\rightarrow S^+_{n+1}$ is the universal covering. Let $G^+_{n+1}$ be the stabilizer of $\tilde{S}^+_{n+1}$ in $G_{n+1}.$

By the construction, $S'_n$ and $S^+_{n+1} $ are 