Malnormal Special Quotient Theorem and results of Bergeron and Wise for the cubulation of groups. Taken together with Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds. Simple loop conjecture If is a map of closed connected surfaces such that is not injective, then there exists a non-contractible simple closed curve such that is homotopically trivial. This conjecture was proven by David Gabai. Surface subgroup conjecture The surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Problem 3.75 in Robion Kirby's problem list. Assuming the geometrization conjecture, the only open case was that of closed hyperbolic 3-manifolds. A proof of this case was announced in the Summer of 2009 by Jeremy Kahn and Vladimir Markovic and outlined in a talk August 4, 2009 at the FRG (Focused Research Group) Conference hosted by the University of Utah. A preprint appeared on the arxiv in October 2009. Their paper was published in the Annals of Mathematics in 2012. In June 2012, Kahn and Markovic were given the Clay Research Awards by the Clay Mathematics Institute at a ceremony in Oxford. Important conjectures Cabling conjecture The cabling conjecture states that if Dehn surgery on a knot in the 3-sphere yields a reducible 3-manifold, then that knot is a -cable on some other knot, and the surgery must have been performed using the slope . Lubotzky–Sarnak conjecture The fundamental group of any finite volume hyperbolic n-manifold does not have Property τ. References Further reading External links Strickland, Neil, A Bestiary of Topological Objects Low-dimensional topology Geometric topology