Hanna Neumann conjecture: Difference between revisions

Latest revision as of 23:42, 7 May 2008

This article is about a conjecture in the following area in/related to group theory: presentation theory. View all conjectures and open problems

This conjecture was made/proposed by: Hanna Neumann

Statement

Let $H$ and $K$ be finitely generated nontrivial subgroups of a free group. Then:

$r a n k (H \cap K) - 1 \leq (r a n k (H) - 1) (r a n k (K) - 1)$

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 $H$ and $K$ be finitely generated nontrivial subgroups of a free group. Then:

$r a n k (H \cap K) - 1 \leq 2 (r a n k (H) - 1) (r a n k (K) - 1)$

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

Offshoots

Myasnikov has raised the following question:

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

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

@@ Line 20: / Line 20: @@
 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==