On the Profinite Topology on Negatively Curved Groups

Rita Gitik

Let $H$ and $K$ be quasiconvex subgroups of a negatively curved LERF group $G$. It is shown that if $H$ is malnormal in $G$ then the double coset $KH$ is closed in the profinite topology of $G$. In particular, this is true if $G$ is the fundamental group of an atoroidal LERF hyperbolic $3$-manifold $M$, and $H$ is the fundamental group of a totally geodesic boundary component $N$ of $M$.