I want to understand the geometry of the Hyperbolic annuli (Hyperbolic plane quoteinted by the group generated by $z\mapsto rz$ for a fixed $r$) and parabolic annuli (Hyperbolic plane quoteinted by the group generated by $z\mapsto z + 1$). Some questions i want to understand are :Injectivity radius at an arbitrary point ; How the shortest geodesic point passing through a point looks? etc.. Can Somebody suggest a good reference?

Thanks in advance.