Property (QT) for 3-manifold groups

TL;DR

This paper characterizes when the fundamental groups of compact, orientable 3-manifolds have property (QT), linking it to the geometric decomposition of the manifold, and extends the property to certain classes of groups.

Contribution

It provides a complete characterization of property (QT) for 3-manifold groups based on their geometric decomposition and establishes property (QT) for specific classes of groups.

Findings

3-manifold groups have property (QT) iff no Sol or Nil geometry summand.

All irreducible 3-manifold groups with nontrivial torus decomposition and no Sol support have property (QT).

Property (QT) holds for Croke-Kleiner admissible and certain relatively hyperbolic groups.

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 $\pi_1(M)$ 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).