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.

中文翻译：

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相对双曲的几乎特殊群的有限完备性

我们根据有限完备性给出了相对双曲实际上特殊群的表征。我们还证明了相对双曲近紧特殊群G的有限完成Ĝ的子群的 Tits 替代方案，并完全描述了Ĝ的有限生成的 Pro -p子群。这适用于双曲算术流形的基本群的有限完备性。我们推断，SO ( n , 1)标准算术格的同余核的所有有限生成的 prop p子群都是自由 prop p。