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The profinite completion of relatively hyperbolic virtually special groups

Pavel Zalesskii

TL;DR

This paper develops a profinite perspective on relatively hyperbolic virtually compact special groups and standard arithmetic hyperbolic manifolds. It shows that toral relative hyperbolicity and relative hyperbolicity with abelian parabolics can be detected algebraically via the profinite completion, and proves a Tits-alternative for closed subgroups of the profinite completion, classifying finitely generated pro-$p$ subgroups as pro-$p$ fundamental groups of finite graphs of pro-$p$ groups. Applying these structural results to standard arithmetic lattices, it provides a detailed description of finitely generated pro-$p$ subgroups in the profinite completions and in the congruence kernel, including that such subgroups are free pro-$p$ in many cases. The findings unify geometric group theory with profinite techniques and yield concrete consequences for arithmetic hyperbolic manifolds, notably the freeness of pro-$p$ subgroups in congruence kernels.

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-$p$.

The profinite completion of relatively hyperbolic virtually special groups

TL;DR

This paper develops a profinite perspective on relatively hyperbolic virtually compact special groups and standard arithmetic hyperbolic manifolds. It shows that toral relative hyperbolicity and relative hyperbolicity with abelian parabolics can be detected algebraically via the profinite completion, and proves a Tits-alternative for closed subgroups of the profinite completion, classifying finitely generated pro- subgroups as pro- fundamental groups of finite graphs of pro- groups. Applying these structural results to standard arithmetic lattices, it provides a detailed description of finitely generated pro- subgroups in the profinite completions and in the congruence kernel, including that such subgroups are free pro- in many cases. The findings unify geometric group theory with profinite techniques and yield concrete consequences for arithmetic hyperbolic manifolds, notably the freeness of pro- subgroups in congruence kernels.

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 of a relatively hyperbolic virtually compact special group and completely describe finitely generated pro- subgroups of . This applies to the profinite completion of the fundamental group of a hyperbolic arithmetic manifold. We deduce that all finitely generated pro- subgroups of the congruence kernel of a standard arithmetic lattice of are free pro-.
Paper Structure (4 sections, 11 theorems, 6 equations)

This paper contains 4 sections, 11 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pro-$p$ subgroups of profinite completions of relatively hyperbolic virtually special groups
  4. Arithmetic hyperbolic manifolds

Key Result

Theorem 1.1

Let $G$ be a virtually compact special group and $p$ be a prime. Then $G$ is hyperbolic if and only if $\widehat{G}$ does not contain $\mathbb{Z}_p\times \mathbb{Z}_p$, where $\mathbb{Z}_p$ is the group of $p$-adic integers.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Theorem 2.4
  • ...and 11 more