Hanna Neumann conjecture

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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]

Contents

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 ≤ 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.

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 a1,...,an 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

s ≤ 2mn − 3m − 2n + 4.

See also

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References

  1. 1 2 Hanna Neumann. On the intersection of finitely generated free groups. Addendum. Publicationes Mathematicae Debrecen, vol. 5 (1957), p. 128
  2. 1 2 Joel Friedman, "Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: With an Appendix by Warren Dicks" Mem. Amer. Math. Soc., 233 (2015), no. 1100.
  3. 1 2 Igor Minevev, "Submultiplicativity and the Hanna Neumann Conjecture." Ann. of Math., 175 (2012), no. 1, 393-414.
  4. ↑ Andrei Jaikin-Zapirain, Approximation by subgroups of finite index and the Hanna Neumann conjecture, Duke Mathematical Journal, 166 (2017), no. 10, pp. 1955-1987
  5. ↑ A. G. Howson. On the intersection of finitely generated free groups. Journal of the London Mathematical Society, vol. 29 (1954), pp. 428–434
  6. ↑ Hanna Neumann. On the intersection of finitely generated free groups. Publicationes Mathematicae Debrecen, vol. 4 (1956), 186–189.
  7. 1 2 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
  8. ↑ Robert G. Burns. On the intersection of finitely generated subgroups of a free group. Mathematische Zeitschrift, vol. 119 (1971), pp. 121–130.
  9. ↑ Gábor Tardos. On the intersection of subgroups of a free group. Inventiones Mathematicae, vol. 108 (1992), no. 1, pp. 29–36.
  10. ↑ John R. Stallings. Topology of finite graphs. Inventiones Mathematicae, vol. 71 (1983), no. 3, pp. 551–565
  11. ↑ Warren Dicks. Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture. Inventiones Mathematicae, vol. 117 (1994), no. 3, pp. 373–389
  12. ↑ G. N. Arzhantseva. A property of subgroups of infinite index in a free group Proc. Amer. Math. Soc. 128 (2000), 3205–3210.
  13. ↑ 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
  14. ↑ 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
  15. ↑ 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.
  16. ↑ 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
  17. ↑ 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
  18. ↑ 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
  19. ↑ 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
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