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Property (QT) for 3-manifold groups

Suzhen Han, Hoang Thanh Nguyen, Wenyuan Yang

TL;DR

This paper characterizes property $QT$ for finitely generated 3-manifold groups by linking group actions on products of quasi-trees to the geometric structure of the manifold. It develops a comprehensive framework around Croke–Kleiner admissible (CKA) groups and relatively hyperbolic groups, establishing $QT$ under natural hypotheses such as omnipotent hyperbolic quotients and full profinite separability of peripheral subgroups. The authors prove that $\pi_1(M)$ has $QT$ precisely when no sphere–disc summand of $M$ supports Sol or Nil geometry, with direct consequences for graph and mixed 3-manifolds. The work combines projection-complex techniques, coned-off hyperbolic geometries, and sophisticated distance formulas to produce explicit embeddings into finite products of quasi-trees, yielding a broad spectrum of applications in geometric group theory and 3-manifold topology.

Abstract

According to Bestvina-Bromberg-Fujiwara, a finitely generated group is said to have property (QT) if it acts isometrically on a finite product of quasi-trees so that orbital maps are quasi-isometric embeddings. We prove that the fundamental group $π_1(M)$ of a compact, connected, orientable 3-manifold $M$ has property (QT) if and only if no summand in the sphere-disc decomposition of $M$ supports either Sol or Nil geometry. In particular, all compact, orientable, irreducible 3-manifold groups with nontrivial torus decomposition and not supporting Sol geometry have property (QT). In the course of our study, we establish property (QT) for the class of Croke-Kleiner admissible groups and of relatively hyperbolic groups under natural assumptions has property (QT).

Property (QT) for 3-manifold groups

TL;DR

This paper characterizes property for finitely generated 3-manifold groups by linking group actions on products of quasi-trees to the geometric structure of the manifold. It develops a comprehensive framework around Croke–Kleiner admissible (CKA) groups and relatively hyperbolic groups, establishing under natural hypotheses such as omnipotent hyperbolic quotients and full profinite separability of peripheral subgroups. The authors prove that has precisely when no sphere–disc summand of supports Sol or Nil geometry, with direct consequences for graph and mixed 3-manifolds. The work combines projection-complex techniques, coned-off hyperbolic geometries, and sophisticated distance formulas to produce explicit embeddings into finite products of quasi-trees, yielding a broad spectrum of applications in geometric group theory and 3-manifold topology.

Abstract

According to Bestvina-Bromberg-Fujiwara, a finitely generated group is said to have property (QT) if it acts isometrically on a finite product of quasi-trees so that orbital maps are quasi-isometric embeddings. We prove that the fundamental group of a compact, connected, orientable 3-manifold has property (QT) if and only if no summand in the sphere-disc decomposition of supports either Sol or Nil geometry. In particular, all compact, orientable, irreducible 3-manifold groups with nontrivial torus decomposition and not supporting Sol geometry have property (QT). In the course of our study, we establish property (QT) for the class of Croke-Kleiner admissible groups and of relatively hyperbolic groups under natural assumptions has property (QT).

Paper Structure

This paper contains 40 sections, 47 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Property (QT) of 3-manifold groups
  4. Property (QT) of Croke-Kleiner admissible groups.
  5. Property (QT) of relatively hyperbolic groups.
  6. Preliminary
  7. Property (QT)
  8. Sub-exponential growth and solvable groups with property (QT)
  9. CKA groups
  10. Strips in CKA spaces
  11. Special paths in CKA spaces
  12. Projection axioms
  13. Property (QT) of relatively hyperbolic groups
  14. Thick distance formula
  15. Proof of property (QT) of relatively hyperbolic groups
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $M$ be a connected, compact, orientable 3-manifold. Then $\pi_1(M)$ has property (QT) if and only if no summand in its sphere-disk decomposition supports either $Sol$ or $Nil$ geometry.

Figures (2)

  • Figure 1: The dotted and blue path from $x$ to $y$ is a special path, and the red path is one $L^1$-version of it.
  • Figure 2: Verification of Axiom 2

Theorems & Definitions (106)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Definition 2.1
  • ...and 96 more