Hanna Neumann conjecture - Revision history

Vipul: 2 revisions

2008-05-07T23:42:54Z

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Vipul: /* Progress towards the conjecture */

2007-06-07T15:27:38Z

Progress towards the conjecture

← Older revision		Revision as of 15:27, 7 June 2007
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	She then asked whether the factor of 2 can be dropped.		She then asked whether the factor of 2 can be dropped.

			==Offshoots==

			Myasnikov has raised the following question:

			Let <math>m</math>, <math>n</math> be positive integers, and <math>H</math> and <math>K</math> nontrivial finitely generated subgroups of a free group such that <math>rank(H) = n</math> and <math>rank(K) = m</math>. Which numbers between 1 and <math>(n-1)(m-1)</math> can be realized as <math>rank(H \cap K) - 1</math>? In particular, can <math>(n-1)(m-1) - 1</math> be realized?

	==References==		==References==

Vipul at 15:21, 7 June 2007

2007-06-07T15:21:18Z

New page

{{conjecture|presentation theory}}

{{conjecture proposed by|Hanna Neumann}}

==Statement==

Let <math>H</math> and <math>K</math> be [[finitely generated group|finitely generated]] nontrivial subgroups of a free group. Then:

<math>rank(H \cap K) - 1 \le (rank(H) - 1 )(rank(K) - 1)</math>

==Progress towards the conjecture==

===Neumann's own work===

Hanna Neumann, in her paper ''On the intersection of finitely generated free groups'', proved the following:

Let <math>H</math> and <math>K</math> be [[finitely generated group|finitely generated]] nontrivial subgroups of a free group. Then:

<math>rank(H \cap K) - 1 \le 2(rank(H) - 1 )(rank(K) - 1)</math>

She then asked whether the factor of 2 can be dropped.

==References==

* ''On the intersection of finitely generated free groups'' by Hanna Neumann, ''Publ. Math. Debrecen'' 4 (1955-56) 186-169
* ''On the intersection of finitely generated subgroups of a free group'' by Robert G. Burns, ''Math Z.'' 119 (1971), 121-130
* ''On intersections of finitely generated subgroups of free groups'' by Walter D. Neumann, ''Groups-Canberra 1989'', ''Lecture Notes in Mathematics Vol. 1456''