The profinite completion of relatively hyperbolic virtually special groups

Abstract: We give a characterization of toral relatively hyperbolic virtually special groups in terms of the profinite completion. We also prove a Tits alternative for subgroups of the profinite completion $\hat G$ of a relatively hyperbolic virtually compact special group $G$ and completely describe finitely generated pro-$p$ subgroups of $\hat G$. This applies to the profinite completion of the fundamental group of a hyperbolic arithmetic manifold. We deduce that all finitely generated pro-$p$ subgroups of the congruence kernel of a standard arithmetic lattice of $SO(n,1)$ are free pro-$p$.