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A notion of quasiconvex subgroups in acylindrically hyperbolic groups

Ping Wan

TL;DR

This work develops a robust notion of quasiconvexity for finitely generated groups endowed with hyperbolically embedded subgroups, showing that $(G,\mathcal{P})$-quasiconvex subgroups admit uniform quasiconvex constants in coned-off cusped spaces $\\mathbb{K}_r$. The authors build a detailed framework of coned-off and cusped graphs, angle metrics, and extended pullbacks to establish uniform control across models and under sufficiently long Dehn fillings. They prove that this quasiconvexity is preserved under fillings and obtain a Groves–Manning type theorem in this broader setting, enabling applications to groups acting on CAT(0) cube complexes. The results yield stable geometric behavior of quasiconvex subgroups under quotient operations and provide tools for transferring convexity properties through Dehn fillings, with implications for projections of geodesics and cube complex quotients.

Abstract

In this paper, we present a notion of quasiconvexity in the setting of finitely-generated groups with hyperbolically embedded subgroups. Our main result shows that this notion yields uniform quasiconvex constants in the setting of coned-off cusped spaces. We also prove that this notion of quasiconvexity is preserved under sufficiently long Dehn filling. As an application, we generalize a theorem of Groves-Manning on groups acting on $\operatorname{CAT}(0)$ cube complexes.

A notion of quasiconvex subgroups in acylindrically hyperbolic groups

TL;DR

This work develops a robust notion of quasiconvexity for finitely generated groups endowed with hyperbolically embedded subgroups, showing that -quasiconvex subgroups admit uniform quasiconvex constants in coned-off cusped spaces . The authors build a detailed framework of coned-off and cusped graphs, angle metrics, and extended pullbacks to establish uniform control across models and under sufficiently long Dehn fillings. They prove that this quasiconvexity is preserved under fillings and obtain a Groves–Manning type theorem in this broader setting, enabling applications to groups acting on CAT(0) cube complexes. The results yield stable geometric behavior of quasiconvex subgroups under quotient operations and provide tools for transferring convexity properties through Dehn fillings, with implications for projections of geodesics and cube complex quotients.

Abstract

In this paper, we present a notion of quasiconvexity in the setting of finitely-generated groups with hyperbolically embedded subgroups. Our main result shows that this notion yields uniform quasiconvex constants in the setting of coned-off cusped spaces. We also prove that this notion of quasiconvexity is preserved under sufficiently long Dehn filling. As an application, we generalize a theorem of Groves-Manning on groups acting on cube complexes.

Paper Structure

This paper contains 21 sections, 55 theorems, 59 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolically Embedded Subgroups
  4. Thick Graphs
  5. Coned-off Cusped Cayley Graph
  6. Dehn Fillings
  7. Quasiconvexity
  8. Quasiconvexity with Coned-off Cayley graph
  9. Angles
  10. Basic Constructions
  11. The Big Angle Property of Coned-off Space
  12. Interplay between Different Graphs
  13. Back and Forth
  14. Angle in Coned-off Cusped Cayley Graph
  15. Extended pullback
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $G$ be a finitely generated group, $\mathcal{P}$ a finite family of infinite subgroups that are hyperbolically embedded into $G$. Let $H$ be a finitely generated subgroup of $G$. If $H$ is $( G,\mathcal{P} )$--quasiconvex, then there exists a subset $Y\subseteq H$, a finite collection of infinit Moreover, there exists $r_\lambda, \lambda$ such that $\forall r\geq r_\lambda$, $\phi_r(\mathbb{K}

Figures (6)

  • Figure 1: A path between $v_1$ and $v_2$ that does not pass through $a$
  • Figure 2: When $l(\widehat{\gamma}_1)\geq 3\delta$, $\angle_{\widehat{x}}(\widehat{\gamma}_1, \widehat{\gamma}_2) < 8\delta$
  • Figure 3: The path that shows $\angle_{\widehat{a} }(\widehat{a}_1, \widehat{a}_2) \leq 8 \delta$.
  • Figure 4: The geodesic triangle in the proof of \ref{['lem:ext-pullback_close_to_geodesic']}
  • Figure 5: A geodesic between $\widehat{\overline x}'$ and $\widehat{\overline y}'$ in $\widehat{K}$
  • ...and 1 more figures

Theorems & Definitions (131)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: dahmaniHyperbolicallyEmbeddedSubgroups2017
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Coned-off Cayley Graph
  • Definition 2.6: martinez-pedrozaNoteHyperbolicallyEmbedded2022
  • Definition 2.7: martinez-pedrozaNoteHyperbolicallyEmbedded2022
  • ...and 121 more