The profinite completion of relatively hyperbolic virtually special groups (Q6561669)

In geometric group theory, a problem of great interest is detecting properties of a group \(G\) and in particular of the fundamental group \(\pi_{1}M\) of a manifold, via its finite quotients, or more conceptually, by its profinite completion \(\widehat{G}\) and \(\widehat{\pi_{1}M}\).\N\NIn the paper under review, the author gives a characterization of toral relatively hyperbolic virtually special groups in terms of the profinite completion. He also proves a Tits alternative for subgroups of the profinite completion \(\widehat{G}\) of a relatively hyperbolic virtually compact special group \(G\) and completely describes finitely generated pro-\(p\) subgroups of \(\widehat{G}\). These results apply to the profinite completion of the fundamental group of a hyperbolic arithmetic manifold. He deduces that all finitely generated pro-\(p\) subgroups of the congruence kernel of a standard arithmetic lattice of \(\mathrm{SO}(n,1)\) are free pro-\(p\). (Standard arithmetic subgroups of \(\mathrm{SO}(n,1)\) are virtually compact special by a result of \textit{N. Bergeron} et al. [J. Lond. Math. Soc., II. Ser. 83, No. 2, 431--448 (2011; Zbl 1236.57021)].)