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

TL;DR

This paper proves that finitely generated groups with linear divergence have a well-defined QR-boundary consisting of a single point, and that all finitely generated 3-manifold groups also have well-defined QR-boundaries.

Contribution

It establishes the existence and structure of QR-boundaries for groups with linear divergence and for all finitely generated 3-manifold groups, expanding understanding of their large-scale geometry.

Findings

QR-boundary is a single point for groups with linear divergence

All finitely generated 3-manifold groups have well-defined QR-boundaries

Advances the understanding of boundaries in geometric group theory

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.