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I seem to remember written or said somewhere that at some point Thurston decided to stop writing down his theorems in order not to repel mathematicians from his field (maybe this is not correct?). I am really curious if now 25-30 years later there is some nice source, book, or notes, where it is possible to learn some basic ideas about the proof of the fact that Haken manifolds admit a hyperbolic structure? Maybe some of his ideas got a more accessible explanation? Of course his beautiful notes http://www.msri.org/publications/books/gt3m/ exist, but they don't go so far.

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From reading the article linked in the answer of HW, one sees that the statement of what the OP remembers having read or heard, that "Thurston decided to stop writing down his theorems in order not to repel mathematicians from his field", is indeed not correct. –  Lee Mosher Mar 16 '13 at 15:29

2 Answers 2

up vote 11 down vote accepted

There are several sources for Thurston's hyperbolization theorem, some published, some not.

Off the top of my head:

1) M.Kapovich, Hyperbolic manifolds and discrete groups.

2) J. Hubbard's Teichmuller theory volume II (not yet published)

3) J. Morgan, H. Bass (eds). The Smith conjecture. (English) Papers presented at the symposium held at Columbia University, New York, 1979. Pure and Applied Mathematics, 112. Academic Press, Inc., Orlando, Fla., 1984. xv+243 pp.

For only the case of manifolds that fibre over S^1

1) J-P. Otal, The hyperbolization theorem for fibred 3-manifolds.

Of course there's also the new non-Thurston proofs using Ricci flow.

Oh, and regarding that anecdote about repelling people from a field -- I've only heard that comment attributed to one mathematician and it was in reference to Thurston's early work on foliations. I don't think that's a widely held belief, but I wasn't alive then so I'm just going on 2nd hand comments.

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There's also MR1677888 (2000b:57025) Otal, Jean-Pierre(F-ENSLY-PM) Thurston's hyperbolization of Haken manifolds. Surveys in differential geometry, Vol. III (Cambridge, MA, 1996), 77--194, Int. Press, Boston, MA, 1998. This takes the approach of McMullen (one could also cite McMullen's papers, but Otal's article puts more of the proof together). –  Ian Agol Dec 4 '09 at 1:38
Really? Thurston himself says something about repelling people from the field here: ams.org/bull/1994-30-02/S0273-0979-1994-00502-6/… –  Qiaochu Yuan Dec 4 '09 at 1:42
Hmm, that Otal book seems to be out of print. :( –  Ryan Budney Dec 4 '09 at 2:20
And an anonymous mathematician reminds me to mention that there is of course all of the relevant papers of Thurston's on the arXiv now. Links: front.math.ucdavis.edu/9801.5019, front.math.ucdavis.edu/9801.5045, front.math.ucdavis.edu/9801.5058 –  Ryan Budney Dec 4 '09 at 5:36
I've scanned a copy of Morgan's book on the Smith Conjecture that I made as a graduate student: dl.dropboxusercontent.com/u/8592391/Smith_conjecture.pdf –  Ian Agol Jan 19 at 18:34

This isn't a direct answer to the actual question, but in your first sentence I think you're alluding to Thurston's article On proof and progress in mathematics. In section 6, entitled "Some personal experiences", he describes how his experience working on foliations influenced the way he presented his later work on geometrization.

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