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.