Surface subgroup conjecture
In mathematics, 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.[1]
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 in the arxiv.org server in October 2009.[2] Their paper was published in the Annals of Mathematics in 2012 .[3] In June 2012, Kahn and Markovic were given the Clay Research Awards by the Clay Mathematics Institute at a ceremony in Oxford.[4]
Notes
- ↑ Robion Kirby, Problems in low-dimensional topology
- ↑ 'Immersing almost geodesic surfaces in a closed hyperbolic three manifold', arXiv:0910.5501
- ↑ 'Immersing almost geodesic surfaces in a closed hyperbolic three manifold',
- ↑ 2012 Clay Research Conference