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.^[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. ^[4]

The subject of the conjecture was originally motivated by a 1954 theorem of Howson^[5] 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^[6] Hanna Neumann improved this bound by showing that:

s − 1 ≤ 2mn − 2m − n.

In a 1957 addendum,^[1] 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.

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

rank(L) − 1 ≤ (rank(H) − 1)(rank(K) − 1).

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.

If H, K ≤ G are two subgroups of a group G and if a, b ∈ G define the same double coset HaK = HbK then the subgroups H ∩ aKa⁻¹ and H ∩ bKb⁻¹ are conjugate in G and thus have the same rank. It is known that if H, K ≤ F(X) are finitely generated subgroups of a finitely generated free group F(X) then there exist at most finitely many double coset classes HaK in F(X) such that H ∩ aKa⁻¹ ≠ {1}. Suppose that at least one such double coset exists and let a₁,...,a_n be all the distinct representatives of such double cosets. The strengthened Hanna Neumann conjecture, formulated by her son Walter Neumann (1990),^[7] states that in this situation

∑ i = 1 n [ r a n k ( H ∩ a i K a i − 1 ) − 1 ] ≤ ( r a n k ( H ) − 1 ) ( r a n k ( K ) − 1 ) . {\displaystyle \sum _{i=1}^{n}[{\rm {rank}}(H\cap a_{i}Ka_{i}^{-1})-1]\leq ({\rm {rank}}(H)-1)({\rm {rank}}(K)-1).}

The strengthened Hanna Neumann conjecture was proved in 2011 by Joel Friedman.^[2] Shortly after, another proof was given by Igor Mineyev.^[3]

• In 1971 Burns improved^[8] Hanna Neumann's 1957 bound and proved that under the same assumptions as in Hanna Neumann's paper one has

s ≤ 2mn − 3m − 2n + 4.

• In a 1990 paper,^[7] Walter Neumann formulated the strengthened Hanna Neumann conjecture (see statement above).
• Tardos (1992)^[9] established the strengthened Hanna Neumann Conjecture for the case where at least one of the subgroups H and K of F(X) has rank two. As most other approaches to the Hanna Neumann conjecture, Tardos used the technique of Stallings subgroup graphs^[10] for analyzing subgroups of free groups and their intersections.
• Warren Dicks (1994)^[11] established the equivalence of the strengthened Hanna Neumann conjecture and a graph-theoretic statement that he called the amalgamated graph conjecture.
• Arzhantseva (2000) proved^[12] that if H is a finitely generated subgroup of infinite index in F(X), then, in a certain statistical meaning, for a generic finitely generated subgroup K {\displaystyle K} in F ( X ) {\displaystyle F(X)} , we have H ∩ gKg⁻¹ = {1} for all g in F. Thus, the strengthened Hanna Neumann conjecture holds for every H and a generic K.
• In 2001 Dicks and Formanek established the strengthened Hanna Neumann conjecture for the case where at least one of the subgroups H and K of F(X) has rank at most three.^[13]
• Khan (2002)^[14] and, independently, Meakin and Weil (2002),^[15] showed that the conclusion of the strengthened Hanna Neumann conjecture holds if one of the subgroups H, K of F(X) is positively generated, that is, generated by a finite set of words that involve only elements of X but not of X⁻¹ as letters.
• Ivanov^[16][17] and Dicks and Ivanov^[18] obtained analogs and generalizations of Hanna Neumann's results for the intersection of subgroups H and K of a free product of several groups.
• Wise (2005) claimed^[19] that the strengthened Hanna Neumann conjecture implies another long-standing group-theoretic conjecture which says that every one-relator group with torsion is coherent (that is, every finitely generated subgroup in such a group is finitely presented).

• Geometric group theory