RMC - Vol. 9, Num. 1 - W. S. Jassim

Full paper in PDF:
$%W. S. Jassim, On the intersection of finitely generated subgroups of free groups, Rev. Mat. Univ. Complut. Madrid 9 (1996), no. 1, 67–84.%$

On the Intersection of Finitely Generated Subgroups of Free Groups

W. S. JASSIM

P.O. Box 4286
Ádhamiyah (Bagdad) — Iraq

Received: June 6, 1994
Revised: April 25, 1995

ABSTRACT

Howson (1954) proved that the intersection of two finitely generated subgroups $H$ and $K$ of ranks $m$ and $n$ respectively is finitely generated. He proved that the rank $N$ of $H /~\ K$ is at most $2mn -m - n +1$ . H. Neumann (1956, 1960) gave a better bound of $2mn -2n -2m + 3$ . Burns (1971) further improved the general upper bound to $N = 2mn - 3m - 2n + 4$ (for $m < n$ ).

Imrich (1977) gave shorter proof of Neumann’s result and also Nickolas (1985) gave a simple proof for Burns’ result. Servatius (1983) gave a graphical proof for Burns’ result.

Burns (1969) showed that the stronger bound $N = mn -n - m +2$ holds if $H$ or $K$ is of finite index in $F$ .

In this paper it is shown that the stronger bound $N = mn -n - m +2$ always holds.

1991 Mathematics Subject Classification: 20E07.