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.



Good references for Hyperbolic and parabolic annuli
Alan Beardon's "Geometry of Discrete Groups" should give you all the tools you need to answer your questions (the tools being computation of distances and angles in a convenient model).


