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.
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]
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 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 |