# Thurston's 24 questions: All settled?

Thurston's 1982 article on three-dimensional manifolds1 ends with $24$ "open questions":

$\cdots$

Two naive questions from an outsider: (1) Have all $24$ now been resolved? (2) If so, were they all resolved in his lifetime?

1Thurston, William P. "Three dimensional manifolds, Kleinian groups and hyperbolic geometry." Bull. Amer. Math. Soc, 6.3 (1982). Also: In Proc. Sympos. Pure Math, vol. 39, pp. 87-111. 1983. Citseer PDF download link.

Answered by Ian Agol, Andy Putman, and Igor Rivin. Ian: "Problems 1-18 have been completely answered....Problems 19-24 are more open-ended," and difficult to declare "all settled" (as emphasized by YCor). But, as Andy says, "with the exception of problem 23." Back to Ian: "One can imagine, however, a complete and satisfactory answer eventually to question 23."

• Each of the 24 questions would be probably worth a separate answer! – YCor Mar 25 '17 at 1:32
• @YCor: You are so right! That deserves an article by a knowledgeable researcher in some journal. I am only wondering (naively) what is the status of the questions. – Joseph O'Rourke Mar 25 '17 at 1:46
• You have to wonder whether 24 was chosen to (literally) one-up Hilbert... – Nate Eldredge Mar 25 '17 at 3:26
• It was no criticism on your post. But sometimes the status of one given question is harder to describe than just "solved" or "unsolved". So I mean that any answer of the form "Here is Question X", here are the progress etc would be welcome. (And btw there are a few MathOF threads that contain more original information than some survey articles by knowledgeable researcher in journals!) – YCor Mar 25 '17 at 4:31
• Please don't edit answers into the question. The question is the question, it's not the first in a series of posts. The answers should be the answer. – Fund Monica's Lawsuit Mar 26 '17 at 6:40

A nice summary of the status of these problems may be found here:

Otal, Jean-Pierre, William P. Thurston: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry", Jahresber. Dtsch. Math.-Ver. 116, No. 1, 3-20 (2014). ZBL1301.00035.

I would charaterize it this way: Problems 1-18 have been completely answered, although some of the answers are still unpublished (but exist as preprints and are still under submission). However, some of these questions (maybe 4., 7, and 8.) are less precisely formulated and somewhat open-ended, so one could argue whether or not they are answered completely. For problem 4., Hodgson's thesis addresses one part of the question: "Describe the limiting geometry which occurs when hyperbolic Dehn surgery breaks down." I could list several other papers on this topic.

Problems 19-24 are more open-ended, and thus will likely never be completely satisfactorily addressed (I discussed some of these problems here). No progress has been made on problem 23 as far as I know (which was due to Milnor originally, not Thurston). One can imagine, however, a complete and satisfactory answer eventually to question 23.

Otal does not comment much on Problem 24, which again is somewhat imprecise (what does "most" mean?). As Igor mentions, Joseph Maher has given a satisfactory answer. One could also argue that this was satisfactorily answered in Hempel's paper together with geometrization (Heegaard distance $>2$ implies hyperbolic). But there is other work on this question, such as giving a model for a hyperbolic manifold of bounded genus and bounded geometry (a lower bound on the injectivity radius) by Brock, Minsky, Namazi, and Souto. Moreover, these authors have a program to understand the unbounded geometry case as well (so it would eventually give a description of the geometry of all hyperbolic manifolds of bounded Heegaard genus in some sense). Thus, one could consider this problem to still be open for a variety of reasons.

They have all been resolved in some fashion with the exception of problem 23, which asserts that the volumes of all hyperbolic 3-manifolds are not rationally related. We know basically nothing about the diophantine properties of hyperbolic volumes.

Yes, and yes, and yes. The 1st by Perelman, the 24th by Joseph Maher.

• That must have been so satisfying to him to have them all resolved! – Joseph O'Rourke Mar 25 '17 at 1:07
• @JosephO'Rourke Hard to tell. Thurston had a certain vision of things, and he had absolutely no doubt that the Geometrization Conjecture (question 1) was true. Moreover, he told me in the late '80s that he believed iit would be settled by curvature flow methods (as, indeed, it was). I am sure he did not have much doubt about question 24 either, although Joseph's first proof was quite complicated and used post-Thurstonian machinery. – Igor Rivin Mar 25 '17 at 1:15
• What about 2 through 23? – zibadawa timmy Mar 25 '17 at 1:22
• @zibadawatimmy Hard to answer, since some of these are somewhat open-ended. – Igor Rivin Mar 25 '17 at 1:52
• Someone downvoted this? Weird. – Igor Rivin Mar 30 '17 at 23:28