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 description Status Year solved 1st Thurston's geometrization conjecture: every 3-manifold can be decomposed into prime manifolds of eight canonical geometries. 2003 2nd Is every finite group action on 3-manifold equivalent to isometric action? 2009 3rd The geometrization conjecture for 3-dimensional orbifolds: if such orbifold have no with no 2-dimesional suborbifolds, can it be geometrically decomposed? 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. 2000–2013 5th Are all Kleinian groups geometrically tame? 1986–1993 6th Can every Kleinian group be obtained as a limit of geometrically finite groups? 2012 7th Develop theory of Schottky groups and their limits, that will be analogous to quasi-Fuchsian groups theory. 2012 8th Analysis of limits of quasi-Fuchsian groups with accidental parabolics. 2000 9th Are all Kleinian groups topologically tame? 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? 2004 11th Ending lamination conjecture: can geometrically tame representations of given group be parametrized by their ending laminations and their parabolics? 2012 12th Describe quasi-isometry type of Kleinian groups 2012 13th Is the limit set of Kleinian groups with Hausdorff dimension less than 2 geometrically finite? 1997 14th Existence of Cannon–Thurston maps for hyperbolic spaces. 2009–2012 15th Is it possible to residually separate finitely-generated subgroups in a finitely-generated Kleinian group? 2013 16th Virtually Haken conjecture: does every aspherical or hyperbolic 3-manifold have a finite Haken cover? 2012 17th Having 3-manifold that is aspherical, does it have finite cover with positive first Betti number? 2013 18th Virtually fibered conjecture: every hyperbolic 3-manifold have a finite cover which is a surface bundle over the circle. 2013 19th Describe topology and geometry of manifolds constructed as quotient spaces of PSL(2,C) by arithmetic subgroups. — 20th Develop software for calculation of canonical form of surface diffeomorphisms and group action of diffeomorphisms of projectivized lamination spaces. 1990s–2000s 21st Develop software to compute hyperbolic structures on 3-manifold. 1990s–2000s 22nd Develop software for tabulation of basics informations about 3-manifolds, ie: their volumes, Chern-Simon invariants or knots. 1990s–2000s 23rd Are hyperbolic volumes of 3-manifold rationally independent? — 24th Existence of hyperbolic structures on 3-manifolds with given Heegaard genus. 2009
See also Geometrization conjecture Hilbert's problems Taniyama's problems List of unsolved problems in mathematics Poincaré conjecture Smale's problems
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