Thurston's 24 questions

American mathematician William Thurston

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.

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 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

See also

References

  1. ^ a b c Thurston, William P. (1982), "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry", Bulletin of the American Mathematical Society, 6 (3): 357–379, doi:10.1090/S0273-0979-1982-15003-0
  2. ^ a b Thurston, William P. (2014), "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry", Jahresbericht der Deutschen Mathematiker, 116: 3–20, doi:10.1365/s13291-014-0079-5