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]

Jeremy Kahn and Vladimir Markovic who first proved the conjecture, Aarhus, 2012.

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.^[2] In June 2012, Kahn and Markovic were given the Clay Research Awards by the Clay Mathematics Institute at a ceremony in Oxford.

See also

• Virtually Haken conjecture
• Ehrenpreis conjecture

References

1. Robion Kirby, Problems in low-dimensional topology
2. Kahn, J.; Markovic, V. (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics. 175 (3): 1127. arXiv:0910.5501. doi:10.4007/annals.2012.175.3.4.