Hyperbolicity and obstructions to PL actions of the circle

We prove the following ``Torus Trichotomy Theorem'': every finitely generated subgroup of $\pls$ is either virtually free, a central extension of a cocompact Fuchsian group by a finite cyclic group, or a one-ended, non-acylindrically-hyperbolic group containing $\mathbb{Z}^2$. In particular, we obta