Question

I seem to remeber written or said somewhere that at some point Thruston decided to stop to write down his theorems in order not to repell 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 strucutre? Maybe some of his ideas got a more accesible explanation? Of course his beautifull notes http://www.msri.org/publications/books/gt3m/ exist, but they don't go so far.

Answer 1

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.

Answer 2

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.