# 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. These questions significantly influenced the development of geometric topology and related fields over the following decades.

## 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. 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.
By 2012, 22 of Thurston's 24 questions had been resolved.

## Table of problems

Thurston's 24 questions are:
| Problem | Brief explanation | Status | Year solved |
| 1st | The geometrization conjecture for 3-manifolds (a generalization of the Poincaré conjecture) | Solved by Grigori Perelman using Ricci flow with surgery | 2003 |
| 2nd | Classification of finite group actions on geometric 3-manifolds | Solved by Meeks, Scott, Dinkelbach, and Leeb | 2009 |
| 3rd | The geometrization conjecture for 3-orbifolds | Solved by Boileau, Leeb, and Porti | 2005 |
| 4th | Global theory of hyperbolic Dehn surgery | 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 | Density of geometrically finite groups | Solved by Namazi-Souto and Ohshika | 2012 |
| 7th | Theory of Schottky groups and their limits | 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 | Solved as consequence of geometric tameness | 2004 |
| 11th | Ending lamination conjecture | Solved by Brock, Canary, and Minsky | 2012 |
| 12th | Describe quasi-isometry type of Kleinian groups | Solved with Ending lamination theorem | 2012 |
| 13th | Hausdorff dimension and geometric finiteness | Solved by Bishop and Jones | 1997 |
| 14th | Existence of Cannon-Thurston maps | Solved by Mahan Mj | 2009-2012 |
| 15th | LERF property for Kleinian groups | Solved by Ian Agol, building on work of Wise | 2013 |
| 16th | Virtually Haken conjecture | Solved by Ian Agol | 2012 |
| 17th | Virtual positive first Betti number | Solved by Ian Agol | 2013 |
| 18th | Virtually fibered conjecture | Solved by Ian Agol | 2013 |
| 19th | Properties of arithmetic subgroups | Unresolved | — |
| 20th | Computer programs and tabulations | Addressed through development of SnapPea and other software | 1990s–2000s |
| 21st | Computer programs and tabulations | Addressed through development of SnapPea and other software | 1990s–2000s |
| 22nd | Computer programs and tabulations | Addressed through development of SnapPea and other software | 1990s–2000s |
| 23rd | Rational independence of hyperbolic volumes | Unresolved | — |
| 24th | Prevalence of hyperbolic structures in manifolds with given Heegaard genus | Solved by Namazi and Souto | 2009 |