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Here I am asking for a reference for the universal cover of hyperbolic Riemann surfaces with geodesic boundaries. For example, I want to know how the universal cover/fundamental domain of hyperbolic Riemann surfaces with boundary look like (what kind of subsets of the Poincare disk are the fundamental domains?) and the hyperbolic structures on surfaces with boundary, the fundamental groups of them (are they Fuchsian groups?), the characterization of elements in the case (for example, characterization of elements of the fundamental group according to the geometry of the surface with boundary, e.g. parabolic elements correspond to cusp, hyperbolic elements correspond to simple closed non-trivial geodesics) etc.

A quick introduction (or reference) from which I can gather enough material within 1 or 2 weeks or so would be highly appreciated !

Thank you very much.

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That depends where you are starting from... And what is 'enough material'? Why are you after this in such a hurry? – David Roberts Jul 23 '12 at 5:44
Search on the topic of ``Fuchsian Groups'', which are properly discontinuous groups of isometries of the hyperbolic plane. Since you seem to express interest only in surfaces (as opposed to orbifolds), you can restrict solely to the torsion free case. Within that topic, learn about cusps, about the limit set of a Fuchsian group, the convex hull of the limit set, and the quotient of the convex hull under the group action. Once you've digested that, and I will make no predictions about how long it will take you to do that, then you will have everything you ask for. – Lee Mosher Jul 23 '12 at 14:05
And if you want more detailed pointers, I suggest looking at the answers to your own earlier question…, or posting on – Lee Mosher Jul 23 '12 at 14:07

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