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If we consider a disk $D$ with $h$ holes and $n$ punctures on the boundary of the disk, then:

  • Is there a uniformization theorem for such surfaces?

  • What is the condition on $h$ and $n$ such that we can consider such surfaces as hyperbolic surfaces?

  • What is the notion of mapping class group? How can we define it?

  • Is there an analog of the Fenchel-Nielsen coordinates for the Teichmuller space $T_{h,n}(D)$ of such surfaces e.g. just hyperbolic length functions?

  • If we attach a compact surface with genus $g$ and $m$ punctures to one of the holes, what would be the answer to the above questions?

  • Should we always resort to the notion of the double of such surfaces to study them?

Useful references are highly appreciated!

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  • $\begingroup$ You might want to look at Decorated Teichmuller Theory by Penner. It looks like he defines this to be the subspace of Teichmuller space of the double invarient under the natural involution. In this case all the questions should be answerable by looking at the double. $\endgroup$
    – user35370
    Commented Mar 26, 2017 at 19:07
  • $\begingroup$ @PaulPlummer, Thank you for the answer. However, is it possible to think about an open Riemann surface without resorting to its double? The doubled surface makes it extremely complicated. Then what condition should I put on the Teichmuller space of the doubled surface to get the Teichmuller space of the original surface? These questions make me feel that the concept of double might not be the right setup to study open Riemann surface. $\endgroup$
    – QGravity
    Commented Apr 22, 2017 at 7:01
  • $\begingroup$ I was actually recently thinking about this question recently, and considering writing a more detailed answer. I am not sure what you restrictions you are thinking are necessary, the above definition is well defined, no restrictions necessary. People have been studying open Riemann surfaces for a while, I am no expert so I just cited a definition I have seen in multiple places, but the "real definition" should be about conformal equivalence, and the punctures, boundary or not, should be considered as marked points. This doubling procedure just brings us to the setting people are use to. $\endgroup$
    – user35370
    Commented Apr 22, 2017 at 15:49
  • $\begingroup$ This recent paper (which is what got me thinking about it again) on the arxiv, I think, implicitly answers all your questions except for F-N coordinates. If I remember correctly (don't have the book on hand) the Penner book discusses that. Basically there is nothing special about boundary punctures, and all reasonable definitions should work (not 100% sure about representation variety definition but I have a couple of guesses on how to get it to work) $\endgroup$
    – user35370
    Commented Apr 22, 2017 at 16:00
  • $\begingroup$ @PaulPlummer, The restriction that I am thinking about is the following: If we use the double of the surface, the Teichmuller space of the original surface is a subset of the Teichmuller space of the double surface. If this is the correct picture, this subset can be obtained using some constriants on the Teichmuller space of the double surface. This is the restriction that I am referring to. $\endgroup$
    – QGravity
    Commented Apr 23, 2017 at 18:52

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