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Do you know a good reference about boundary of hyperbolic spaces (following Gromov) and the classification of the isometries acting on hyperbolic space (hyperbolic, parabolic and elliptic isometries)? I am specially interested by the hyperbolic isometries, with the notions of axis and translation length.

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I don't have a reference at hand, but the boundary of hyperbolic plane is isomorphic to the real projective line. If you know the effect of an isometry on the boundary, you know the transformation. If there are two fixed points on the boundary, you either have a reflection (which you can recognize as an involution) or, depending on order, a translation or a glide reflection. If you have a single fixed point, you have an ideal rotation. If you have no fixed points on the boundary, the fixed points in $\mathbb C\mathrm P^1$ represent a single hyperbolic point, the center of a rotation. Useful? –  MvG Feb 9 '13 at 22:19
Did you try Ghys-de la Harpe or Bridson-Haefliger or Coornaert-Delzant-Papadopoulos? @MvG: The question seems to be about $\delta$-hyperbolic spaces –  Martin Feb 9 '13 at 23:51
@Martin: The book of Coornaert, Delzant and Papadopoulos seems to be exactly what I seek, however it is not available in my university... Do you know an equivalent reference? –  Seirios Feb 10 '13 at 10:01
Sorry, unfortunately I don't. When I learned about Gromov-hyperbolicity in the Old Millennium, the references I gave were pretty much the only introductory books available. I don't know the newer books on the subject. –  Martin Feb 10 '13 at 16:37
@Martin: Finally, the book of Ghys and de la Harpe seems to be nice too. Since you answered my question, I think you may post these references as an answer. –  Seirios Feb 11 '13 at 10:30

2 Answers 2

up vote 3 down vote accepted

I just mention the references given by Martin:

For my question, I found the third reference better.

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My apologies for not following up on this earlier (I didn't see your comment). My colleague didn't have further suggestions, but she pointed out that Buyalo had some related results in Geodesics in Hadamard spaces, Algebra i Analiz, 10:2 (1998), 93–123 (there is an English translation). –  Martin Mar 29 '13 at 10:52
I will simply mention the "American" version of the "French" and "Swiss" versions of the notes mentioned above: cmi.univ-mrs.fr/~hamish/Papers/MSRInotes2004.pdf . –  user641 Mar 29 '13 at 11:02

A very accessible and well-written reference that covers this is Bonahon's book Low dimensional topology: from Euclidean surfaces to hyperbolic knots. There is also the added advantage of the author's magnificent moustache which can be seen on the back cover.

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