Hyperbolicity and obstructions to PL actions of the circle

Abstract

We investigate acylindrically hyperbolic groups acting on the circle by piecewise linear homeomorphisms. We prove that a finitely generated acylindrically hyperbolic group acting faithfully by piecewise linear homeomorphisms of the circle is virtually free, is virtually a closed hyperbolic surface group, or virtually splits over a two-ended subgroup; as a consequence, we prove a general structural result for Gromov hyperbolic groups of piecewise linear homeomorphisms. Along the way, we show that a right-angled Artin subgroup of piecewise linear homeomorphisms of the circle is either free or abelian, generalizing a result of Bleak and Salazar-Diaz. We also show that the fundamental group of a finite volume hyperbolic $n$--manifold cannot act faithfully on the circle by piecewise linear homeomorphisms, provided that $n\geq 3$, complementing $2$--dimensional examples of Ghys and Minakawa.