From: magidin@math.berkeley.edu (Arturo Magidin)
Subject: Re: Newmann's conjecture
Date: 6 Oct 2000 15:05:05 -0500
Newsgroups: sci.math.research
Summary: [missing]

In article <39DAD04F.346C0F54@cict.fr>, Eric Guirbal  <guirbal@cict.fr> wrote:
>
>  A theorem of Howson [1] claims that in a free group the intersection S
>of two finitely generated subgroups S1 and S2 is finitely generated.  We
>have the following Newmann's inequality on the rank of S [2]. If S is
>non trivial, then
>
>
>rank(S)-1<=2(rank(S1)-1)(rank(S2)-1)
>
>H. Newmann's conjecture is that in the above inequality "2" can be
>replaced by "1".

Actually, the name is spelled "Neumann".

>I would like to know if this conjecture is always a conjecture?

Presumably, you mean whether it is still a conjecture...

There are a number of partial results, including a recent
preprint. One paper, published in Spain, claims to have established
the conjecture, but Warren Dicks has published a correction to that
paper. According to the Magnus Project page, they are still
'discussing' it, although the date given there is over a year old. 

I would suggest looking at the Magnus Project page, at

http://zebra.sci.ccny.cuny.edu/web/

and look for it under "Open Problems", subsection "Outstanding
Problems", problem O9.

Also, a recent preprint available from the arXiv is relevant:

"The Hanna Neumann Conjecture is true when one subgroup has a positive
generating set" by Bilal Khan, preprint number GR/0009152 at the
arXiv. You can get there through 
//front.math.ucdavis.edu/

======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@math.berkeley.edu