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Hyperbolically embedded subgroups and quasi-isometries of pairs

Sam Hughes, Eduardo Martínez-Pedroza

TL;DR

This work develops a framework for understanding when acylindrically hyperbolic groups retain hyperbolic features under quasi-isometries by introducing and analyzing quasi-isometries of pairs $(G,\mathcal P) \to (H,\mathcal Q)$. Central to the approach is transferring hyperbolically embedded subgroups via coned-off Cayley graphs and uniform quasi-actions, leading to precise criteria: if $\mathcal P$ and $\mathcal Q$ are reduced, then $\mathcal P \hookrightarrow_h (G,S)$ iff $\mathcal Q \hookrightarrow_h (H,T)$, and refined transfers $\mathcal P^*$ under infinitude of $\mathcal Q$; these results build on and extend qi-characteristic concepts. The paper also demonstrates how these principles behave under finite-index extensions and semidirect products, showing that hyperbolically embedded subgroups can be preserved or realized in larger groups when actions satisfy freeness and invariance conditions. Collectively, the results provide a structural route toqi-invariance and commensurability questions for acylindrically hyperbolic groups, and they yield concrete applications to semidirect products and refinements of subgroup collections.

Abstract

We give technical conditions for a quasi-isometry of pairs to preserve a subgroup being hyperbolically embedded. We consider applications to the quasi-isometry and commensurability invariance of acylindrical hyperbolicity of finitely generated groups.

Hyperbolically embedded subgroups and quasi-isometries of pairs

TL;DR

This work develops a framework for understanding when acylindrically hyperbolic groups retain hyperbolic features under quasi-isometries by introducing and analyzing quasi-isometries of pairs . Central to the approach is transferring hyperbolically embedded subgroups via coned-off Cayley graphs and uniform quasi-actions, leading to precise criteria: if and are reduced, then iff , and refined transfers under infinitude of ; these results build on and extend qi-characteristic concepts. The paper also demonstrates how these principles behave under finite-index extensions and semidirect products, showing that hyperbolically embedded subgroups can be preserved or realized in larger groups when actions satisfy freeness and invariance conditions. Collectively, the results provide a structural route toqi-invariance and commensurability questions for acylindrically hyperbolic groups, and they yield concrete applications to semidirect products and refinements of subgroup collections.

Abstract

We give technical conditions for a quasi-isometry of pairs to preserve a subgroup being hyperbolically embedded. We consider applications to the quasi-isometry and commensurability invariance of acylindrical hyperbolicity of finitely generated groups.
Paper Structure (6 sections, 21 theorems, 30 equations)

This paper contains 6 sections, 21 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Uniform quasi-actions
  3. Quasi-isometries and Hyperbolically embedded subgroups
  4. Hyperbolically embedded subgroups and commensurability
  5. Semi-direct products and Hyperbolically embedded subgroups
  6. Concluding remarks

Key Result

Theorem A

Let $q\colon G\to H$ be a quasi-isometry of finitely generated groups, let ${\mathcal{P}}$ and ${\mathcal{Q}}$ be finite collections of subgroups of $G$ and $H$ respectively, and let $S$ and $T$ be (not necessarily finite) generating sets of $G$ and $H$ respectively. Suppose The following statements hold:

Theorems & Definitions (55)

  • Definition 1.2: Quasi-isometry of pairs
  • Definition 1.3: Quasi-isometry of group pairs
  • Example 1.4: Quasi-isometry of pairs and finite extensions
  • Definition 1.5: Refinements
  • Example 1.6: Refinements and qi of pairs
  • Example 1.7: Refinements and finite extensions
  • Definition 1.8: Reduced collections
  • Theorem A: \ref{['prop:summary']}
  • Definition 1.9: Qi-characteristic
  • Example 1.10
  • ...and 45 more