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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.

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up vote 1 down vote accepted

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).

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