Hanna Neumann conjecture

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:

$rank(H\cap K)-1\le (rank(H)-1)(rank(K)-1)$

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:

$rank(H\cap K)-1\le 2(rank(H)-1)(rank(K)-1)$

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 $rank(H)=n$ and $rank(K)=m$. Which numbers between 1 and $(n-1)(m-1)$ can be realized as $rank(H\cap K)-1$? 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