Variational integrator

2012-09-03T09:34:01Z

78.91.45.155: In References: It's G. Wanner, not Warner

In the mathematical subject of [[group theory]], the '''Hanna Neumann conjecture''' is a statement about the [[rank of a group|rank]] of the intersection of two [[finitely generated group|finitely generated]] [[subgroup]]s of a [[free group]]. The conjecture was posed by [[Hanna Neumann]] in 1957.<ref name="HN57">Hanna Neumann. ''On the intersection of finitely generated free groups. Addendum.'' [[Publicationes Mathematicae Debrecen]], vol. 5 (1957), p. 128</ref> In 2011, the conjecture was proved independently by Igor Mineyev<ref name="proof">Igor Minevev,
[http://annals.math.princeton.edu/2012/175-1/p11/ "Submultiplicativity and the Hanna Neumann Conjecture."]
Ann. of Math., 175 (2012), no. 1, 393-414</ref> and Joel Friedman. <ref name="proof2">Joel Friedman,
[http://www.math.ubc.ca/~jf/pubs/web_stuff/shnc_memoirs.pdf "Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture."]
to appear in Memoirs of the AMS</ref>

==History==

The subject of the conjecture was originally motivated by a 1954 theorem of Howson<ref>A. G. Howson. ''On the intersection of finitely generated free groups.'' [[Journal of the London Mathematical Society]], vol. 29 (1954), pp. 428–434</ref> who proved that the intersection of any two [[finitely generated group|finitely generated]] [[subgroup]]s of a [[free group]] is always finitely generated, that is, has finite [[rank of a group|rank]]. In this paper Howson proved that if ''H'' and ''K'' are [[subgroup]]s 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<ref>Hanna Neumann. ''On the intersection of finitely generated free groups.'' Publicationes Mathematicae Debrecen, vol. 4 (1956), 186–189.</ref> [[Hanna Neumann]] improved this bound by showing that :

:''s'' − 1 ≤ 2''mn'' − ''2m'' − ''n''.

In a 1957 addendum,<ref name="HN57"/> 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 a group|rank]] of ''G'', that is, the smallest size of a [[generating set of a group|generating set]] for ''G''.
Every [[subgroup]] of a [[free group]] is known to be [[free group|free]] itself and the [[rank of a group|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 (mathematics)|group]] ''G'' and if ''a'', ''b'' ∈ ''G'' define the same [[double coset]] ''HaK = HbK'' then the [[subgroup]]s ''H'' ∩ ''aKa''−1 and ''H'' ∩ ''bKb''−1 are [[Conjugacy class|conjugate]] in ''G'' and thus have the same [[rank of a group|rank]]. It is known that if ''H'', ''K'' ≤ ''F''(''X'') are [[finitely generated group|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),<ref name="WN">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</ref> states that in this situation

:<math>\sum_{i=1}^n [{\rm rank}(H\cap a_iKa_{i}^{-1})-1] \le ({\rm rank}(H)-1)({\rm rank}(K)-1).</math>

==Partial results and other generalizations==

*In 1971 Burns improved<ref>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.</ref> 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,<ref name="WN"/> Walter Neumann formulated the strengthened Hanna Neumann conjecture (see statement above).
*[[Gábor Tardos|Tardos]] (1992)<ref>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.</ref> 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 graph]]s<ref>John R. Stallings. [http://www.springerlink.com/content/mn2h645qw2058530/ ''Topology of finite graphs.''] [[Inventiones Mathematicae]], vol. 71 (1983), no. 3, pp. 551–565</ref> for analyzing subgroups of free groups and their intersections.
*Warren Dicks (1994)<ref>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</ref> established the equivalence of the strengthened Hanna Neumann conjecture and a graph-theoretic statement that he called the ''amalgamated graph conjecture''.
*Arzhantseva (2000) proved<ref>G. N. Arzhantseva. [http://www.ams.org/proc/2000-128-11/S0002-9939-00-05508-8/home.html ''A property of subgroups of infinite index in a free group''] [[Proceedings of the American Mathematical Society|Proc. Amer. Math. Soc.]] 128 (2000), 3205–3210.</ref> 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 <math>K</math> in <math>F(X)</math>, 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 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.<ref>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</ref>
*Khan (2002)<ref>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</ref> and, independently, Meakin and Weil (2002),<ref>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.</ref> 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<ref>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</ref><ref>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</ref> and, subsequently, Dicks and Ivanov,<ref>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</ref> obtained analogs and generalizations of Hanna Neumann's results for the intersection of [[subgroup]]s ''H'' and ''K'' of a [[free product]] of several groups.
*Wise (2005) showed<ref>[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</ref> 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 group|finitely generated]] subgroup in such a group is [[finitely presented group|finitely presented]]).

==See also==
*[[Rank of a group]]
*[[Geometric group theory]]

==References==
{{reflist}}

[[Category:Group theory]]
[[Category:Geometric group theory]]

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Variational integrator