Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups

A relatively hyperbolic group $G$ is said to be QCERF if all finitely generated relatively quasiconvex subgroups are closed in the profinite topology on $G$. Assume that $G$ is a QCERF relatively hyperbolic group with double coset separable (e.g., virtually polycyclic) peripheral subgroups. Given any two finitely generated relatively quasiconvex subgroups $Q,R⩽G$ we prove the existence of finite index subgroups $Q^{\mathrm{\prime }}{⩽}_{f}Q$ and $R^{\mathrm{\prime }}{⩽}_{f}R$ such that the join $⟨Q^{\mathrm{\prime }},R^{\mathrm{\prime }}⟩$ is again relatively quasiconvex in $G$. We then show that, under the minimal necessary hypotheses on the peripheral subgroups, products of finitely generated relatively quasiconvex subgroups are closed in the profinite topology on $G$. From this we obtain the separability of products of finitely generated subgroups for several classes of groups, including limit groups, Kleinian groups and balanced fundamental groups of finite graphs of free groups with cyclic edge groups.