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I am asking a soft question here.

I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three books on manifolds and Riemann surfaces by S. Donaldson. Now I am interested in learning more advanced hyperbolic geometry such as Teichmuller Theory (earthquake theorem), 3-manifolds etc., i.e.

  • The relation geometry of 3-dimensional hyperbolic and anti-de Sitter manifolds and geometry of crooked plane

  • The relations between 3-dimensional geometry and Teichmüller Theory

I believe that to learn the early mentioned topics I need to find an advisor. But, as of now, I am planning to learn those on my own. But I am little bit confused about how I should learn this. Here, I am listing some books which I want to study on my own. The lists are as follows.

  1. The Geometry of Discrete Groups by A. Beardon.
  2. Automorphisms of surfaces after Nielsen and Thurston by Casson and Bleiler.
  3. Teichmüller theory I by Hubbard.
  4. Teichmüller theory II by Hubbard
  5. Univalent Functions and Teichmüller Spaces by O. Lehto.
  6. A Primer on Mapping Class Groups" by Farb and Margalit.
  7. Hyperbolic Manifolds and Kleinian Groups by Katsuhiko Matsuzaki and Masahiko Taniguchi
  8. An Introduction to Geometric Topology by Bruno Martelli
  9. The geometry and topology of three-manifolds by William Thurston

I know that I am writing for a long reading project. But I want to start the self- reading project as much as I can. Later, I will look for an advisor who will guide me (also, to find a advisor I should learn some of these topics to help them believe that I am well-prepared to work under his/her research group).

Above I have mentioned some books for hyperbolic geometry. But I don't know in which order I should learn the books. Moreover, I feel that those books are not in right orders. Please advise me how to study those books in orders. Also, it will be nice if you advise me for a learning roadmap for hyperbolic geometry toward the topics such as the relation geometry of 3-dimensional hyperbolic and anti-de Sitter manifolds and the relations between 3-dimensional geometry and Teichmüller theory.

Please advise me. Thanking in advanced.

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    $\begingroup$ This is a vast field with a substantial written and unwritten literature, and I don't think it is possible to give in the abstract a straightforward path though it. This would depend hugely on your background, interests, talents, learning style, etc. In other words, I'm suggesting that you really need an advisor who knows you well to help you out. $\endgroup$
    – Andy Putman
    Commented Apr 5, 2021 at 20:20
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    $\begingroup$ Besides Thurston’s lecture notes, which seem to be the last point in your list, there is also Thurston’s book “Three-dimensional geometry and topology”, which is easier and more detailed (and has less intersection with the lecture notes than one might have thought). $\endgroup$
    – ThiKu
    Commented Apr 5, 2021 at 20:50
  • $\begingroup$ The subject is vast indeed. Many topics are covered in the "Handbook of Teichmüller theory" (volume I through VII !) ems-ph.org/books/… $\endgroup$
    – Nicolast
    Commented Apr 7, 2021 at 20:13

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