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 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

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