Project on limit groups and their subdirect products

### 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
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.