Thurston's 24 questions

Thurston's 24 questions are a set of mathematical problems in differential geometry posed by American mathematician William Thurston in his influential 1982 paper Three-dimensional manifolds, Kleinian groups and hyperbolic geometry published in the Bulletin of the American Mathematical Society.^[1] These questions significantly influenced the development of geometric topology and related fields over the following decades.

American mathematician William Thurston

History

The questions appeared following Thurston's announcement of the geometrization conjecture, which proposed that all compact 3-manifolds could be decomposed into geometric pieces.^[1] This conjecture, later proven by Grigori Perelman in 2003, represented a complete classification of 3-manifolds and included the famous Poincaré conjecture as a special case.^[2]

By 2012, 22 of Thurston's 24 questions had been resolved.^[2]

Table of problems

Thurston's 24 questions are:^[1]

Problem	Brief description	Status	Year solved
1st	Thurston's geometrization conjecture: every 3-manifold can be decomposed into prime manifolds of eight canonical geometries.	Solved by Grigori Perelman using Ricci flow with surgery.	2003
2nd	Is every finite group action on 3-manifold equivalent to isometric action?	Solved by Meeks, Scott, Dinkelbach, and Leeb.	2009
3rd	The geometrization conjecture for 3-dimensional orbifolds: if such orbifold have no with no 2-dimesional suborbifolds, can it be geometrically decomposed?	Solved by Boileau, Leeb, and Porti.	2005
4th	Global theory of hyperbolic Dehn surgery: give upper bound for nonhyperbolic surgeries and find description of geometry that is created when hyperbolic surgery breaks down.	Resolved through work of Agol, Lackenby, and others.	2000–2013
5th	Are all Kleinian groups geometrically tame?	Solved through work of Bonahon and Canary.	1986–1993
6th	Can every Kleinian group be obtained as a limit of geometrically finite groups?	Solved by Namazi-Souto and Ohshika	2012
7th	Develop theory of Schottky groups and their limits, that will be analogous to quasi-Fuchsian groups theory.	Resolved through work of Brock, Canary, and Minsky.	2012
8th	Analysis of limits of quasi-Fuchsian groups with accidental parabolics.	Solved by Anderson and Canary.	2000
9th	Are all Kleinian groups topologically tame?	Solved independently by Agol and by Calegari-Gabai.	2004
10th	The Ahlfors measure zero problem: group obtained as a limit set of finitely-generated Kleinian group have either full measure or measure 0. In case of full measure, does it act ergodically?	Solved as consequence of geometric tameness.	2004
11th	Ending lamination conjecture: can geometrically tame representations of given group be parametrized by their ending laminations and their parabolics?	Solved by Brock, Canary, and Minsky.	2012
12th	Describe quasi-isometry type of Kleinian groups	Solved with ending lamination theorem.	2012
13th	Is the limit set of Kleinian groups with Hausdorff dimension less than 2 geometrically finite?	Solved by Bishop and Jones.	1997
14th	Existence of Cannon–Thurston maps for hyperbolic spaces.	Solved by Mahan Mj.	2009-2012
15th	Is it possible to residually separate finitely-generated subgroups in a finitely-generated Kleinian group?	Solved by Ian Agol, building on work of Wise.	2013
16th	Virtually Haken conjecture: does every aspherical or hyperbolic 3-manifold have a finite Haken cover?	Solved by Ian Agol.	2012
17th	Having 3-manifold that is aspherical, does it have finite cover with positive first Betti number?	Solved by Ian Agol.	2013
18th	Virtually fibered conjecture: every hyperbolic 3-manifold have a finite cover which is a surface bundle over the circle.	Solved by Ian Agol.	2013
19th	Describe topology and geometry of manifolds constructed as quotient spaces of PSL(2,C) by arithmetic subgroups.	Unresolved.	—
20th	Develop software for calculation of canonical form of surface diffeomorphisms and group action of diffeomorphisms of projectivized lamination spaces.	Addressed through development of SnapPea and other software.	1990s–2000s
21st	Develop software to compute hyperbolic structures on 3-manifold.	Addressed through development of SnapPea and other software.	1990s–2000s
22nd	Develop software for tabulation of basics informations about 3-manifolds, ie: their volumes, Chern-Simon invariants or knots.	Addressed through development of SnapPea and other software.	1990s–2000s
23rd	Are hyperbolic volumes of 3-manifold rationally independent?	Unresolved.	—
24th	Existence of hyperbolic structures on 3-manifolds with given Heegaard genus.	Solved by Namazi and Souto.	2009

See also

• Geometrization conjecture
• Hilbert's problems
• Taniyama's problems
• List of unsolved problems in mathematics
• Poincaré conjecture
• Smale's problems