surface subgroup conjecture

File:Jeremy Kahn and Vladimir Markovic.jpg who first proved the conjecture, Aarhus, 2012.]]

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.Robion Kirby, [http://math.berkeley.edu/~kirby/problems.ps.gz Problems in low-dimensional topology]

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. Their paper was published in the Annals of Mathematics in 2012.{{Cite journal | doi = 10.4007/annals.2012.175.3.4 | arxiv = 0910.5501| title = Immersing almost geodesic surfaces in a closed hyperbolic three manifold| journal = Annals of Mathematics| volume = 175| issue = 3| pages = 1127| year = 2012| last1 = Kahn | first1 = J. | last2 = Markovic | first2 = V. | s2cid = 32593851| authorlink2 = Vladimir Markovic}} In June 2012, Kahn and Markovic were given the Clay Research Awards by the Clay Mathematics Institute at a ceremony in Oxford.