History
The subject of the conjecture was originally motivated by a 1954 theorem of Howson 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 ≤ 2mn − m − n.
In a 1956 paper Hanna Neumann improved this bound by showing that :
- s − 1 ≤ 2mn − 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.
Read more about this topic: Hanna Neumann Conjecture
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