Promotion of pseudo-fibering conjecture

Let $G$ be a finitely-generated torsion-free pseudo-fibered group which does not split as a nontrivial free product. Here, a pseudo-fibered group is a subgroup of $\operatorname{Homeo^+}(S^1)$ whose action on $S^1$ admits two invariant, very full, loose laminations with distinct endpoints. The group $G$ is elementary when it is virtually abelian.

Promotion of pseudo-fibering conjecture. There are three possibilities: $G$ is elementary; $G$ is a Fuchsian group; or $G$ is a closed hyperbolic $3$-manifold group.

This conjecture is a proposed characterization of groups arising from fibered hyperbolic $3$-manifolds. The paper explains the necessity of the torsion-free and no-splitting hypotheses and collects evidence for the conjecture, but does not resolve it.