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what are good books on hyperbolic geometry/hyperbolic surfaces that have good number of exercises, just to get a good understanding of the literature . I know John Ratcliffe's book will be one of them, but it is kind of encyclopedic. I particularly want to solve some exercises where they will be using free homotopy, isotopy, Arzela-Ascoli arguments, covering spaces, lifting of curves and homotopies, and will be actually using hyperbolicity of the manifold.

Thanks !

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    $\begingroup$ Outer Circles by Albert Marden has a ton of exercises and hints. $\endgroup$ – j.c. Sep 30 '10 at 0:34
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Here are some:

Stillwell, "Geometry of Surfaces."

Bonahon, "Low-dimensional geometry. From Euclidean surfaces to hyperbolic knots."

Katok, "Fuchsian groups."

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  • $\begingroup$ Jurgen Jost's "Compact Riemann Surfaces" would be another. $\endgroup$ – Ryan Budney Sep 29 '10 at 20:43
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    $\begingroup$ and Hubbard's "Teichmueller Theory Vol. 1" $\endgroup$ – Ryan Budney Sep 29 '10 at 21:20
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I warmly recommand

Francis Bonahon, "Low-dimensional geometry: from euclidean surfaces to hyperbolic knots"

Many illustrations, many exercises (as far as I remember). Maybe a bit too elementary for what you ask.

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Al Marden's Outer circles is overflowing with excellent exercises. Although the majority of the book is about 3-manifolds, the first two chapters are an introduction to hyperbolic geometry brimming with vim.

Beardon's Geometry of Discrete Groups, Iversen's Hyperbolic Geometry, and Bonahon's Low-dimensional Geometry, and Katok's Fuchsian Groups all have exercises. Since you requested stuff specifically on surfaces, Katok may be the way to go. However, you may want to turn to other books for explanations at times, for this book is terse. Iversen has an entire chapter on covering spaces that you might want to look at.

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Another book to consider is Casson & Bleiler's "Automorphisms of Surfaces after Nielsen and Thurston". While there may not be many explicitly stated exercises, there are many sketches of proofs where working out the details would make very nice exercises for a motivated student (especially if it were a reading course). However, if the student is at an earlier stage, I would second Autumn Kent's recommendation: Stillwell's "Geometry of Surfaces" which is very nice and approachable for an undergraduate student.

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I'm surprised no one's mentioned William Thurston's classic "Three Dimensional Geometry and Topology",which has a significant amount of coverage on hyperbolic geometry in 3-space. Stillwell is better for beginners,though.

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    $\begingroup$ Thurston's book has a much broader mandate, that's probably the reason people had not mentioned it yet. Also, the "step length" can be quite large. $\endgroup$ – Ryan Budney Sep 29 '10 at 21:22
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John Stillwell's book The Geometry of Surfaces is best if you are an undergraduate; if you are a more advanced than that, or an advanced undergraduate, Thurston's book Three Dimensional Geometry and Topology is the best. Either Katok's Fuchsian groups or Hubbard's new Teichmueller Theory Vol. 1, would be reasonable alternatives to Thurston; or, even better, supplemental reading to Thurston on weekend nights with some wine.

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