The profinite completion of relatively hyperbolic virtually special groups

TL;DR

This paper characterizes toral relatively hyperbolic virtually special groups via their profinite completions, establishes a Tits alternative for subgroups of these completions, and describes finitely generated pro-p subgroups, with applications to hyperbolic arithmetic manifolds.

Contribution

It provides a new characterization of certain hyperbolic groups through their profinite completions and describes the structure of their pro-p subgroups, extending understanding of their algebraic properties.

Findings

Profinite completion characterizes toral relatively hyperbolic virtually special groups.

A Tits alternative is established for subgroups of the profinite completion.

Finitely generated pro-p subgroups of the congruence kernel are free pro-p.

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-$p$.