Is it possible to write the infinite jungle gym surface as the increasing union of compact surfaces whose boundaries consist of only one simple closed curve? I believe that this is true. This surface has only one end. Therefore the result follows from the existence of canonical exhaustions (made of compact surfaces whose components of the complement have one boundary component) (as explained in Ahlfors-Sario book). If anyone could help me, I would like to "see" such a curve. Thanks.
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2$\begingroup$ The answer to this question is "yes". This is an example of an "infinite Loch Ness surface"; see arxiv.org/pdf/1701.07151.pdf for a discussion of these surfaces and references to the classification of surfaces of infinite type. All infinite-genus $1$ ended surfaces are diffeomorphic, and thus are diffeomorphic to the canonical example of the union of the genus-$g$ surfaces $\Sigma_{g,1}$ with one boundary component as $g$ goes to infinity (the boundary curves here give you the curves you are looking for). I'm not a good artist, so I'll leave others to give you explicit pictures. $\endgroup$– RobertaCommented Mar 21, 2021 at 3:51
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$\begingroup$ (note: I was not the person who downvoted, and in fact haven't registered my account and thus do not have voting rights at all!) $\endgroup$– RobertaCommented Mar 21, 2021 at 4:15
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1$\begingroup$ Consider an exhaustion of this surface by an $n \times n \times n$ "cube" (for each $n$), with $6(n+1)^2$ boundary circles coming from all the "rods" sticking out. Now draw a giant circuituous circle (simple closed curve) on the surface that encloses all these boundary circles. $\endgroup$– Kevin CastoCommented Mar 21, 2021 at 5:57
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1$\begingroup$ Cross-posted here. In general, you should not post a questions simultaneously here and at MSE. $\endgroup$– Moishe KohanCommented Mar 23, 2021 at 18:37
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