On Infinitely generated Fuchsian groups of some infinite genus surfaces

In this paper, for a non compact and orientable surface $S$ been either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we construct explicitly an infinitely generated Fuchsian group $\mathrm{\Gamma }<PSL(2,\mathbb{R})$, such that the quotient $\mathbb{H}\mathrm{/}\mathrm{\Gamma }$ is a hyperbolic surface homeomorphic to $S$.