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Quasi-redirecting boundaries of groups with linear divergence and 3-manifold groups

Hoang Thanh Nguyen

TL;DR

The paper develops the quasi-redirecting boundary $\partial_{*}$ as a robust, quasi-isometry invariant boundary generalizing the Gromov boundary. It establishes that a finitely generated group with linear divergence has a well-defined, single-point QR-boundary and proves that every finitely generated 3-manifold group has a well-defined QR-boundary. The approach relies on a precise QR-framework with Assumptions 0–2, analysis of quasi-geodesic rays, and a divergence-focused argument to construct redirecting paths. The results unify the QR-boundary behavior across broad classes, including Sol and Nil geometries, and highlight a connection between divergence and the structure of QR-boundaries, with strong implications for coarse geometry of 3-manifold groups.

Abstract

The quasi-redirecting (QR) boundary, introduced by Qing and Rafi, generalizes the Gromov boundary for studying the large-scale geometry of finitely generated groups. Although it is not known to exist for all such groups, its existence has been established for several important classes. We prove that if a finitely generated group G has linear divergence, its QR-boundary is well-defined and consists of a single point. In addition, we show that all finitely generated 3-manifold groups admit well-defined QR-boundaries.

Quasi-redirecting boundaries of groups with linear divergence and 3-manifold groups

TL;DR

The paper develops the quasi-redirecting boundary as a robust, quasi-isometry invariant boundary generalizing the Gromov boundary. It establishes that a finitely generated group with linear divergence has a well-defined, single-point QR-boundary and proves that every finitely generated 3-manifold group has a well-defined QR-boundary. The approach relies on a precise QR-framework with Assumptions 0–2, analysis of quasi-geodesic rays, and a divergence-focused argument to construct redirecting paths. The results unify the QR-boundary behavior across broad classes, including Sol and Nil geometries, and highlight a connection between divergence and the structure of QR-boundaries, with strong implications for coarse geometry of 3-manifold groups.

Abstract

The quasi-redirecting (QR) boundary, introduced by Qing and Rafi, generalizes the Gromov boundary for studying the large-scale geometry of finitely generated groups. Although it is not known to exist for all such groups, its existence has been established for several important classes. We prove that if a finitely generated group G has linear divergence, its QR-boundary is well-defined and consists of a single point. In addition, we show that all finitely generated 3-manifold groups admit well-defined QR-boundaries.
Paper Structure (7 sections, 12 theorems, 27 equations, 2 figures)

This paper contains 7 sections, 12 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Coarse geometry
  4. QR-Assumptions
  5. Divergence of groups
  6. Quasi-redirecting boundaries of groups with linear divergence
  7. Quasi-redirecting boundary of finitely generated 3-manifold groups

Key Result

Theorem 1.4

Let $G$ be a finitely generated group. If $G$ has linear divergence, then the QR-boundary $\partial_{*} G$ consists of exactly one point.

Figures (2)

  • Figure 1: The cacatenation $\alpha \cup [\alpha_{+}, p] \cup \gamma$ is a quasi-geodesic.
  • Figure 2: The figure demonstrates the concatenation $\gamma : = [o, p']_{\alpha} \cup p', a] \cup [ a, b] \cup [b, d]_{\sigma} \cup [d, e] \cup \bar{\beta}|_{[e, \infty)}$ is the desired quasi-geodesic which quasi-redirects $\alpha$ to $\bar{\beta}$ at radius $\frac{35s}{72}$

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.4
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: QR24
  • Definition 2.7
  • ...and 17 more