Quasigeodesic flows and Möbius-like groups

Abstract

If M is a hyperbolic 3-manifold with a quasigeodesic flow, then we show that $\pi_1(M)$ acts in a natural way on a closed disc by homeomorphisms. Consequently, such a flow either has a closed orbit or the action on the boundary circle is Möbius-like but not conjugate into $PSL(2,\mathbb{R})$.We conjecture that the latter possibility cannot occur.