Property (QT) for 3-manifold groups

Let M be a compact, connected, orientable 3-manifold with no summand supporting SL (2, R) geometry in its sphere-disc decomposition. 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 ₁ (M) has property (QT) if and only if M does not contain Sol and Nil geometries. 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 classes of Croke-Kleiner admissible groups and of relatively hyperbolic groups under natural assumptions. Accordingly, this yields that graph 3-manifold and mixed 3-manifold groups have property (QT). The question whether the SL (2, R) lattices have property (QT) is left open.