Project on limit groups and their subdirect products
Martin R. Bridson and James Howie
Paper 1: Normalizers in limit groups
Preprint, May 2005; minor modifications July 2006
--- to appear in Math. Ann.
- This is a .pdf file
(10 pages, no figures)
Let $\G$ be a limit group,
$S\subset\G$ a non-trivial subgroup, and $N$ the normaliser
of $S$.
If $H_1(S,\mathbb Q)$ has
finite $\Q$-dimension, then $S$ is finitely generated
and
either $N/S$ is finite or $N$ is abelian.
This result has applications to the
study of subdirect
products of limit groups.
Paper 2: Subgroups of direct products of elementarily free groups
Preprint, June 2005 --- to appear in GAFA.
- This is a .pdf file
(19 pages, no figures)
We exploit Zlil Sela's description of
the structure of groups having the same
elementary theory as free groups:
they and their
finitely generated subgroups form a prescribed
subclass $\mathcal E$ of the hyperbolic limit groups.
We prove that if $G_1,\dots,G_n$ are in $\mathcal E$
then a subgroup $\Gamma\subset G_1\times\dots\times G_n$
is of type $\FP_n$ if and only if $\Gamma$ is
itself, up to finite index,
the direct product of at most
$n$ groups from $\mathcal E$. This
answers a question of Sela.
Paper 3: Subgroups of direct products of two limit groups
Preprint, October 2005 --- submitted for publication.
- This is a .pdf file
(18 pages, no figures)
If $\G_1$ and $\G_2$ are limit
groups and $S\subset \G_1\times\G_2$ is of type $\FP_2$
then $S$ has a subgroup of finite index that
is a product of at most two limit groups.