Problem
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Brief description
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Status
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Year solved
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1st
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Thurston's geometrization conjecture: every 3-manifold can be decomposed into prime manifolds of eight canonical geometries.
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Solved by Grigori Perelman using Ricci flow with surgery.
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2003
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2nd
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Is every finite group action on 3-manifold equivalent to isometric action?
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Solved by Meeks, Scott, Dinkelbach, and Leeb.
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2009
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3rd
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The geometrization conjecture for 3-dimensional orbifolds: if such orbifold have no with no 2-dimesional suborbifolds, can it be geometrically decomposed?
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Solved by Boileau, Leeb, and Porti.
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2005
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4th
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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.
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Resolved through work of Agol, Lackenby, and others.
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2000–2013
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5th
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Are all Kleinian groups geometrically tame?
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Solved through work of Bonahon and Canary.
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1986–1993
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6th
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Can every Kleinian group be obtained as a limit of geometrically finite groups?
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Solved by Namazi-Souto and Ohshika
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2012
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7th
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Develop theory of Schottky groups and their limits, that will be analogous to quasi-Fuchsian groups theory.
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Resolved through work of Brock, Canary, and Minsky.
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2012
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8th
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Analysis of limits of quasi-Fuchsian groups with accidental parabolics.
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Solved by Anderson and Canary.
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2000
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9th
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Are all Kleinian groups topologically tame?
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Solved independently by Agol and by Calegari-Gabai.
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2004
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10th
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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?
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Solved as consequence of geometric tameness.
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2004
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11th
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Ending lamination conjecture: can geometrically tame representations of given group be parametrized by their ending laminations and their parabolics?
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Solved by Brock, Canary, and Minsky.
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2012
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12th
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Describe quasi-isometry type of Kleinian groups
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Solved with ending lamination theorem.
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2012
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13th
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Is the limit set of Kleinian groups with Hausdorff dimension less than 2 geometrically finite?
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Solved by Bishop and Jones.
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1997
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14th
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Existence of Cannon–Thurston maps for hyperbolic spaces.
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Solved by Mahan Mj.
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2009–2012
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15th
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Is it possible to residually separate finitely-generated subgroups in a finitely-generated Kleinian group?
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Solved by Ian Agol, building on work of Wise.
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2013
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16th
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Virtually Haken conjecture: does every aspherical or hyperbolic 3-manifold have a finite Haken cover?
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Solved by Ian Agol.
|
2012
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17th
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Having 3-manifold that is aspherical, does it have finite cover with positive first Betti number?
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Solved by Ian Agol.
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2013
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18th
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Virtually fibered conjecture: every hyperbolic 3-manifold have a finite cover which is a surface bundle over the circle.
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Solved by Ian Agol.
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2013
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19th
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Describe topology and geometry of manifolds constructed as quotient spaces of PSL(2,C) by arithmetic subgroups.
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Unresolved.
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—
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20th
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Develop software for calculation of canonical form of surface diffeomorphisms and group action of diffeomorphisms of projectivized lamination spaces.
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Addressed through development of SnapPea and other software.
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1990s–2000s
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21st
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Develop software to compute hyperbolic structures on 3-manifold.
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Addressed through development of SnapPea and other software.
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1990s–2000s
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22nd
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Develop software for tabulation of basics informations about 3-manifolds, ie: their volumes, Chern-Simon invariants or knots.
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Addressed through development of SnapPea and other software.
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1990s–2000s
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23rd
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Are hyperbolic volumes of 3-manifold rationally independent?
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Unresolved.
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—
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24th
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Existence of hyperbolic structures on 3-manifolds with given Heegaard genus.
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Solved by Namazi and Souto.
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2009
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