# 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]

## History

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 ≤ 2*mn*−*m*−*n*.

In a 1956 paper[6] Hanna Neumann improved this bound by showing that :

*s*− 1 ≤ 2*mn*−*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*.

## Formal statement

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.

## Strengthened Hanna Neumann conjecture

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 *a*_{1},...,*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

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

## Partial results and other generalizations

- 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*≤ 2*mn*− 3*m*− 2*n*+ 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 in , 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).

## See also

## References

- Hanna Neumann.
*On the intersection of finitely generated free groups. Addendum.*Publicationes Mathematicae Debrecen, vol. 5 (1957), p. 128 - Joel Friedman, "Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture" American Mathematical Soc., 2014
- Igor Minevev, "Submultiplicativity and the Hanna Neumann Conjecture." Ann. of Math., 175 (2012), no. 1, 393-414.
- Andrei Jaikin-Zapirain,
*Approximation by subgroups of finite index and the Hanna Neumann conjecture*, Duke Mathematical Journal,**166**(2017), no. 10, pp. 1955-1987 - A. G. Howson.
*On the intersection of finitely generated free groups.*Journal of the London Mathematical Society, vol. 29 (1954), pp. 428–434 - Hanna Neumann.
*On the intersection of finitely generated free groups.*Publicationes Mathematicae Debrecen, vol. 4 (1956), 186–189. - Walter Neumann.
*On intersections of finitely generated subgroups of free groups.*Groups–Canberra 1989, pp. 161–170. Lecture Notes in Mathematics, vol. 1456, Springer, Berlin, 1990; ISBN 3-540-53475-X - Robert G. Burns.
*On the intersection of finitely generated subgroups of a free group.*Mathematische Zeitschrift, vol. 119 (1971), pp. 121–130. - Gábor Tardos.
*On the intersection of subgroups of a free group.*Inventiones Mathematicae, vol. 108 (1992), no. 1, pp. 29–36. - John R. Stallings.
*Topology of finite graphs.*Inventiones Mathematicae, vol. 71 (1983), no. 3, pp. 551–565 - Warren Dicks.
*Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture.*Inventiones Mathematicae, vol. 117 (1994), no. 3, pp. 373–389 - G. N. Arzhantseva.
*A property of subgroups of infinite index in a free group*Proc. Amer. Math. Soc. 128 (2000), 3205–3210. - Warren Dicks, and Edward Formanek.
*The rank three case of the Hanna Neumann conjecture.*Journal of Group Theory, vol. 4 (2001), no. 2, pp. 113–151 - Bilal Khan.
*Positively generated subgroups of free groups and the Hanna Neumann conjecture.*Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), 155–170, Contemporary Mathematics, vol. 296, American Mathematical Society, Providence, RI, 2002; ISBN 0-8218-2822-3 - J. Meakin, and P. Weil.
*Subgroups of free groups: a contribution to the Hanna Neumann conjecture.*Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000). Geometriae Dedicata, vol. 94 (2002), pp. 33–43. - S. V. Ivanov.
*Intersecting free subgroups in free products of groups.*International Journal of Algebra and Computation, vol. 11 (2001), no. 3, pp. 281–290 - S. V. Ivanov.
*On the Kurosh rank of the intersection of subgroups in free products of groups*. Advances in Mathematics, vol. 218 (2008), no. 2, pp. 465–484 - Warren Dicks, and S. V. Ivanov.
*On the intersection of free subgroups in free products of groups.*Mathematical Proceedings of the Cambridge Philosophical Society, vol. 144 (2008), no. 3, pp. 511–534 -
*The Coherence of One-Relator Groups with Torsion and the Hanna Neumann Conjecture.*Bulletin of the London Mathematical Society, vol. 37 (2005), no. 5, pp. 697–705