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In the paper: "CONSTRUCTING PAIRS OF PANTS" by Ara Basmajian (Source: http://www.acadsci.fi/mathematica/Vol15/vol15pp065-074.pdf), the author defined:

A torsion free Fuchsian group $G$ is said to be a pair of pants if $H^2/G$ is topologically a sphere with 3 holes. The pair of paints is tight if one of the holes is really a puncture (that is, an open neighborhood of it is hyperbolically a punctured disc).

I knew that a pair of pants with geodesic boundary cannot be a quotient space of $H^2$ by some Fuchsian group because their universal cover is a proper subset of $H^2$. So I guess that with the above definition, we can construct the only pair of pants with 3 cusps. If my guess is wrong, could someone please give me some counterexamples of Fuchsian groups so that the quotient space is a pair of pants with geodesic boundary. Thanks in advance!

  • Basmajian is defining what it means for the group $G$ to a pair of pants. Topologists define what it means for a topological surface to be a pair of pants. Geometers define what it means for a hyperbolic surface to be pair of pants. You are mixing these all up together and noticing that they are not perfectly consistent with each other, which is not a big surprise. I advise learning to keep the contexts clear, and make sure you know which definition is used in which context. – Lee Mosher Feb 27 '18 at 12:50
  • Thanks for your comment. I understand what you meant so I added one more sentence from his paper where he wanted to separate special case when the hole becomes a cusp. –  Feb 27 '18 at 13:06

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The key concept that you are still missing, and which helps tie together all the different contexts, is that a geometric pair of pants can have one of four possibilities: 3 cusps; 2 cusps and 1 geodesic boundary; 1 cusp and 2 geodesic boundaries; or 3 geodesic boundaries. Furthermore, whichever of these is appropriate can be "constructed" from the Fuchsian group $G$.

The point is that the "construction" is not just as simple as taking the quotient of the whole of $H^2$ with respect to the $G$-action, as your question pre-supposes. Instead, the correct construction is to take the quotient of the convex hull of the limit set with respect to the $G$-action.

For example, if the group $G$ has no parabolic elements then the quotient of the convex hull of the limit set with respect to the $G$-action will be a geometric pair of pants with 3 geodesic boundary components. More generally, the number of cusps of the corresponding geometric pair of pants equals the number of conjugacy classes in $G$ of maximal infinite cyclic parabolic subgroups.

Lee Mosher
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    Thank you so much for the answer. I think " convex hull of the limit set" is the keywords that I need. –  Feb 27 '18 at 13:42