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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, (''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 . In June 2012, Kahn and Markovic were given the Clay Research Awards by the Clay Mathematics Institute at a ceremony in Oxford.〔(2012 Clay Research Conference )〕 ==See also== *Virtually Haken conjecture *Ehrenpreis conjecture 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「surface subgroup conjecture」の詳細全文を読む スポンサード リンク
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