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 Ĝ of a relatively hyperbolic virtually compact special group G and completely describe finitely generated pro-p subgroups of Ĝ. 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.