  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).
