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In the mathematical subject of group theory, the Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957.〔Hanna Neumann. ''On the intersection of finitely generated free groups. Addendum.'' Publicationes Mathematicae Debrecen, vol. 5 (1957), p. 128〕 In 2011, the conjecture was proved independently by Joel Friedman 〔Joel Friedman, ("Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture." ) American Mathematical Soc., 2014.〕 and Igor Mineyev.〔Igor Minevev, ("Submultiplicativity and the Hanna Neumann Conjecture." ) Ann. of Math., 175 (2012), no. 1, 393-414〕 ==History== The subject of the conjecture was originally motivated by a 1954 theorem of Howson〔A. G. Howson. ''On the intersection of finitely generated free groups.'' Journal of the London Mathematical Society, vol. 29 (1954), pp. 428–434〕 who proved that the intersection of any two finitely generated subgroups of a free group is always finitely generated, that is, has finite rank. In this paper Howson proved that if ''H'' and ''K'' are subgroups of a free group ''F''(''X'') of finite ranks ''n'' ≥ 1 and ''m'' ≥ 1 then the rank ''s'' of ''H'' ∩ ''K'' satisfies: :''s'' − 1 ≤ 2''mn'' − ''m'' − ''n''. In a 1956 paper〔Hanna Neumann. ''On the intersection of finitely generated free groups.'' Publicationes Mathematicae Debrecen, vol. 4 (1956), 186–189.〕 Hanna Neumann improved this bound by showing that : :''s'' − 1 ≤ 2''mn'' − ''2m'' − ''n''. In a 1957 addendum,〔 Hanna Neumann further improved this bound to show that under the above assumptions :''s'' − 1 ≤ 2(''m'' − 1)(''n'' − 1). She also conjectured that the factor of 2 in the above inequality is not necessary and that one always has :''s'' − 1 ≤ (''m'' − 1)(''n'' − 1). This statement became known as the ''Hanna Neumann 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「Hanna Neumann conjecture」の詳細全文を読む スポンサード リンク
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