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I want to read hyperbolic geometry. Can any one suggest some good books on the topic.

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  • $\begingroup$ What is your background? $\endgroup$ – user38268 Jan 10 '13 at 11:13
  • $\begingroup$ I have knowledge in complex analysis, basic functional analysis, topology,basic algebra $\endgroup$ – Phani Raj Jan 10 '13 at 11:17
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A professor of mine suggested Euclidean and Non-Euclidean Geometries: Development and History by Marvin J. Greenberg when I asked him the same question.

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There's 56-page introductory paper on hyperbolic geometry by Cannon, Floyd, Kenyon, and Parry which is absolutely excellent. You can download it from Kenyon's web page:

Cannon, Floyd, Kenyon, and Parry, Hyperbolic Geometry.

Note that this paper is much more directed towards the modern point of view than most sources, and is not at all interested in synthetic (or axiomatic) geometry.

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Geometry and Topology of Three-Manifolds by Bill Thurston, edited by Silvio Levy.

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I can recommend Low-Dimensional Geometry by Francis Bonahon and Chapter 2 of Thurston's Three-Dimensional Geometry and Topology (ed. Levy).]

You could go on to Al Marden's Outer Circles or Benedetti and Petronio's Lectures on Hyperbolic Geometry if that whets your appetite.

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You may enjoy Chapter 6 of Needham's Visual Complex Analysis.

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In Cannon, Floyd, Kenyon, and Parry, Hyperbolic Geometry, the authors recommend:

[Iversen 1993]for starters, and [Benedetti and Petronio 1992; Thurston 1997; Ratcliffe 1994] for more advanced readers. The latter has a particularly comprehensive bibliography.

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A bit of a fanciful introduction is "Journey into Geometries" (Amazon.com link) by Marta Sved and H. S. M. Coxeter. It's a narrative exploration in (more-or-less) the style of Lewis Carroll's Wonderland stories.

From the Amazon description:

This unique book gives an informal introduction into the non-Euclidean geometries through a series of dialogues between a somewhat grown-up Alice (of Looking Glass fame), her uncle Lewis Carroll, and a visitor from the twentieth century, Dr Whatif. In the story, Lewis Carroll's geometrical beliefs are cast into the Euclidean mould, Dr Whatif asks the penetrating and controversial questions, and Alice acts as a mediator and interested participant. The book is intentionally more mathematical than Lewis Carroll's books, but for those of us who enjoyed Alice's earlier adventures there are many interesting flashbacks to those inimitable characters: the Red Queen, Tweedle-Dum and his twin brother, the Mad Hatter ... The text is filled with humour, wit, and verses of poetry. Part 1 contains the story in six chapters, each of which concludes with a problem set; Part 2 is more mathematical, and looks at the axiom systems, and gives solutions to the problems. The presentation, with its old-time borders, script headings, and cartoon drawings evokes the spirit of the original Alice.

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    $\begingroup$ This simple short book is indeed a topnotch recommendation and it is a must to read. Nobody studies geometry anymore. I had to comment on it. It is for those who care. $\endgroup$ – Ken Draco Aug 1 at 7:49
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There's a short one called "About Lobachevski's Geometry" By Smogorzhevski. I translated the name from the Spanish copy I have, so it could be called differently. Sadly, that's the only one I know, because I haven't studied a lot about this geometry, and all I know was from general or euclidean geometry books that had a couple of chapters about introduction to non-euclidean geometries. That book is for beginners though, so with the background you say you have, you won't have any trouble at all with that.

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  • $\begingroup$ I think you mean the booklet (70 pages only) A.S. Smogorzhevski called "Lobachevskian Geometry" published by MIR Publishers Moscow. It is a nice booklet, but to short to be a good introduction $\endgroup$ – Willemien Aug 30 '15 at 20:43

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