On Quasiconvex Subgroups of Negatively Curved Groups

Rita Gitik 

MSRI #1995-024

We say that a group is locally quasiconvex if for some finite
presentation all its finitely generated subgroups are quasiconvex.
Let $G$ and $H$ be locally quasiconvex subgroups of a negatively
curved group of $GH$ (the amalgamated prodcut of $G$ and $H$ with
$G_0=H_0$) and let $L$ be a finitely generated subgroup of $GH$ which
has finitely generated intersection with any conjugate of $G$ or of
$H$.  We prove that if $G_0$ is malnormal in $G$ and quasiconvex in
$GH$ then $L$ is quasiconvex in $A$.

In particularly, a free product of  locally quasiconvex negatively curved 
groups is locally quasiconvex  and  a free product of two 
negatively curved locally quasiconvex groups amalgamated over
a  cyclic subgroup which is malnormal in one of the factors is locally 
quasiconvex.

We also give a new  proof of the fact that  
locally quasiconvex groups have the finitely generated intersection property,
hence the groups mentioned above have the
finitely generated intersection property.