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A Characterization of Relative Hyperbolicity via Morse and Contracting Boundaries

Vyshnav PT, Pranab Sardar, Rana Sardar

Abstract

We prove the following boundary-theoretic characterization of relatively hyperbolic groups. Let $G$ be a finitely generated group with a finite collection $\mathcal{H}$ of finitely generated subgroups, and let $G^h$ denote the associated cusped space. We prove that the pair $(G,\mathcal{H})$ is non-elementary relatively hyperbolic if and only if the Morse boundary $\partial_M^{\mathcal{DL}} G^h$ or the contracting boundary $\partial_c^{\mathcal{FQ}} G^h$ is non-empty and compact.

A Characterization of Relative Hyperbolicity via Morse and Contracting Boundaries

Abstract

We prove the following boundary-theoretic characterization of relatively hyperbolic groups. Let be a finitely generated group with a finite collection of finitely generated subgroups, and let denote the associated cusped space. We prove that the pair is non-elementary relatively hyperbolic if and only if the Morse boundary or the contracting boundary is non-empty and compact.
Paper Structure (8 sections, 19 theorems, 23 equations)

This paper contains 8 sections, 19 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Morse Boundary and Contracting Boundary
  3. Morse Boundary
  4. Contracting Boundary
  5. Stability of Weak Hull
  6. Pal-Pandey Characterizations of Relative Hyperbolicity
  7. Relatively hyperbolic groups
  8. Proof of Theorem \ref{['thm_1']}

Key Result

Theorem 1.1

Suppose $G$ is a finitely generated group that is not virtually cyclic and $\mathcal{H}$ is a collection of finitely generated, infinite index, infinite subgroups of $G$. Let $G^h$ denote the cusped space associated to the pair $(G,\mathcal{H})$. Then the following are equivalent:

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1: Morse (quasi) geodesic
  • Lemma 2.2: Morse Quasi-Geodesic Stability, cordes2017
  • Lemma 2.3: cordes2017
  • Lemma 2.4: cordes2017, Cordes2024
  • Lemma 2.5: Cordes2024
  • Definition 2.6: Cordes2024
  • Lemma 2.7: cordes2019, cordes2017
  • Definition 2.8: Contracting quasi-geodesic
  • Theorem 2.9: ACGH2017, CM2019
  • ...and 19 more