Hyperbolicity and obstructions to PL actions of the circle

Abstract

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 obtain the ``free or surface'' alternative for word-hyperbolic groups of PL homeomorphisms of the circle.

49 pages. The main theorem has been strengthened