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In this paper, for a non-compact Riemman surface  S  homeomorphic to either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we give a precise description of an infinite set of generators of a Fuchsian group Γ &lt;  PSL (2, ℝ), such that the quotient space ℍ/Γ is a hyperbolic Riemann surface homeomorphic to  S . For each one of these constructions, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

On Infinitely generated Fuchsian groups of the Loch Ness monster, the Cantor tree and the Blooming Cantor tree