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.
[1] In 2011, a strengthened version of the conjecture (see below) was proved independently by Joel Friedman[2] and by Igor Mineyev.
[3] In 2017, a third proof of the Strengthened Hanna Neumann conjecture, based on homological arguments inspired by pro-p-group considerations, was published by Andrei Jaikin-Zapirain.
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: In a 1956 paper[6] Hanna Neumann improved this bound by showing that: In a 1957 addendum,[1] Hanna Neumann further improved this bound to show that under the above assumptions She also conjectured that the factor of 2 in the above inequality is not necessary and that one always has This statement became known as the Hanna Neumann conjecture.
The strengthened Hanna Neumann conjecture, formulated by her son Walter Neumann (1990),[7] states that in this situation The strengthened Hanna Neumann conjecture was proved in 2011 by Joel Friedman.