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 1957Hanna Neumann. "On the intersection of finitely generated free groups. Addendum." Publicationes Mathematicae Debrecen, vol. 5 (1957), p. 128 ] and still remains open.

History

The subject of the conjecture was originally motivated by a 1954 theorem of Howson [A. G. Howson. "On the intersection of finitely generated free groups." Journal of the London Mathematical Society, vol. 29 (1954), pp. 428–434 ] 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 a 1956 paper [Hanna Neumann. "On the intersection of finitely generated free groups." Publicationes Mathematicae Debrecen, vol. 4 (1956), 186–189. ] Hanna Neumann gave a quantitavie version of Hauson's result and 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 1957 addendum, Hanna Neumann 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" 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.

trengthened 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 Walter Neumann (1990)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] , states that in this situation

:$sum\_\{i=1\}^n\; [\{\; m\; rank\}(Hcap\; a\_iKa\_\{i\}^\{-1\})-1]\; le\; (\{\; m\; rank\}(H)-1)(\{\; m\; rank\}(K)-1).$

Partial results and other generalizations

*In 1971 Burns improved [Robert G. Burns. [http://www.springerlink.com/content/u75hh74lu1053790/ "On the intersection of finitely generated subgroups of a free group."]
Mathematische Zeitschrift, vol. 119 (1971), pp. 121–130.] 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, Walter Neumann formulated the strengthened Hanna Neumann conjecture (see statement above).
*Tardos (1992) [Gábor Tardos. [http://www.springerlink.com/content/n013g5rx543x4748/ "On the intersection of subgroups of a free group."]
Inventiones Mathematicae, vol. 108 (1992), no. 1, pp. 29–36.] established the 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 [John R. Stallings. [http://www.springerlink.com/content/mn2h645qw2058530/ "Topology of finite graphs."] Inventiones Mathematicae, vol. 71 (1983), no. 3, pp. 551–565] for analyzing subgroups of free groups and their intersections.
*Warren Dicks (1994) [Warren Dicks. [http://www.springerlink.com/content/r526373840056u7q/ "Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture."] Inventiones Mathematicae, vol. 117 (1994), no. 3, pp. 373–389 ] established the equivalence of the strengthened Hanna Neumann conjecture and a graph-theoretic statement that he called the "amalgamated graph conjecture".
*In 2001 Dicks and Formanek used this equivalence to prove the strengthened Hanna Neumann Conjecture in the case when one of the subgroups "H" and "K" of "F"("X") has rank at most three. [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]
*Khan (2002) [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] and, independently, Meakin and Weil (2002) [J. Meakin, and P. Weil. [http://www.springerlink.com/content/m742547j1g534g40/ "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.] , 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 [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 ] and, subsequently, Dicks and Ivanov [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] , 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) showed [ [http://blms.oxfordjournals.org.proxy2.library.uiuc.edu/cgi/content/abstract/37/5/697 "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] 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).

ee also

*Free group
*Rank of a group
*Geometric group theory

References