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

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 Gamma < PSL(2, R), such that the quotient space H/Gamma 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.