A month ago Aougab, Patel and Vlamis posted a preprint on the arXiv (arXiv:2007.01982) about the question which groups, for a fixed orientable surface of infinite genus, can be realized as the full isometry group of a Riemannian metric of constant negative curvature on that surface.
To my surprise, they stated in the introduction that it is already known that every countable group can be realized as the full isometry group of some complete hyperbolic surface, citing both Allcock (https://www.jstor.org/stable/4098165) and Winkelmann (https://www.math.uni-bielefeld.de/documenta/vol-06/16.pdf) for this result.
I had a look into both papers, i.e., Allcock’s and Winkelmann’s, and was surprised that they are very short (3 and 6 pages, respectively). So my original goal of posting here the idea of the proof of this fact was ruined since you can, if you are interested, just as easily read Allcock’s paper (the actual proof is just a single page and very elementary: you just glue pairs of pants suitably together).
And just as a side remark: Allcock’s paper – though proving the same result as Winkelmann’s one, but with a shorter, more elementary and easier to understand proof – was published in a worse journal than Winkelmann’s paper.