Hanna Neumann conjecture

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. In 2011, a strengthened version of the conjecture (see below) was proved by Joel Friedman who posted his proof on arXiv on May 1, 2011. Shortly after, another proof was given by Igor Mineyev who posted his proof on his personal web page on May 6, 2011.

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:

In a 1956 paperHanna Neumann improved this bound by showing that :

In a 1957 addendum, 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.

Let H, K ≤ F(X) be two nontrivial finitely generated subgroups of a free group F(X) and let L = H ∩ K be the intersection of H and K. The conjecture says that in this case

Here for a group G the quantity rank(G) is the rank of G, that is, the smallest size of a generating set for G. Every subgroup of a free group is known to be free itself and the rank of a free group is equal to the size of any free basis of that free group.

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