Universal circles for quasigeodesic flows

Abstract

We show that if $M$ is a hyperbolic $3$–manifold which admits a quasigeodesic flow, then ${\pi }_1\left(M\right)$ acts faithfully on a universal circle by homeomorphisms, and preserves a pair of invariant laminations of this circle. As a corollary, we show that the Thurston norm can be characterized by quasigeodesic flows, thereby generalizing a theorem of Mosher, and we give the first example of a closed hyperbolic $3$–manifold without a quasigeodesic flow, answering a long-standing question of Thurston.