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.

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 Thurston's geometrization conjecture for 3-manifolds (a generalization of the Poincaré conjecture). Solved by Grigori Perelman using Ricci flow with surgery. 2003
2nd Is every finite group action on a 3-manifold equivalent to an action respecting the geometry? Solved by Meeks, Scott, Dinkelbach, and Leeb 2009
3rd Does every 3-dimensional orbifold which contains no bad 2-dimensional suborbifolds admit a geometric decomposition? 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 Is every Kleinian group a limit of geometrically finite groups? Solved by Namazi-Souto and Ohshika 2012
7th Develop a theory of Schottky groups and their limits analogous to the theory of quasi-fuchsian 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: Does the limit set of a finitely-generated Kleinian group always have full measure or 0 measure, and in the former case does group act ergodically? Solved as consequence of geometric tameness 2004
11th Ending lamination conjecture: are geometrically tame representations of a given group parametrized by their ending laminations and their parabolics, together with the conformai structure on the domain of discontinuity? Solved by Brock, Canary, and Minsky 2012
12th Describe the quasi-isometry type of an arbitrary Kleinian group. Solved with Ending lamination theorem 2012
13th If the limit set of a finitely-generated Kleinian group has Hausdorff dimension less than 2, is it geometrically finite? Solved by Bishop and Jones 1997
14th Existence of Cannon–Thurston maps. Solved by Mahan Mj 2009-2012
15th Can finitely-generated subgroups of a finitely-generated Kleinian group be residually separated from the group? Solved by Ian Agol, building on work of Wise 2013
16th Virtually Haken conjecture: Does every aspherical 3-manifold, or every hyperbolic 3-manifold, have a finite-sheeted cover which is Haken? Solved by Ian Agol 2012
17th Does every aspherical 3-manifold have a finite-sheeted cover with positive first Betti number? Solved by Ian Agol 2013
18th Virtually fibered conjecture: Does every hyperbolic 3-manifold have a finite-sheeted cover which fibers over the circle? Solved by Ian Agol 2013
19th Find topological and geometric properties of quotient spaces of arithmetic subgroups of . Unresolved —
20th Develop a computer program to calculate the canonical form for a general diffeomorphism of a surface, and to calculate the action of the group of diffeomorphisms . Addressed through development of SnapPea and other software 1990s–2000s
21st Develop a computer program to calculate hyperbolic structures on 3-manifolds. Addressed through development of SnapPea and other software 1990s–2000s
22nd Tabulate the volumes and the Chern-Simons invariants and other simple information for a bunch of 3-manifolds. Addressed through development of SnapPea and other software 1990s–2000s
23rd Show that the volumes of hyperbolic 3-manifolds are not all rationally related. Unresolved —
24th Show that most 3-manifolds with Heegard diagrams of a given genus have hyperbolic structures. 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
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