Quasigeodesic flows and M\"obius-like groups

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 M"obius-like but not conjugate into PSL(2, R). We conjecture that the latter possibility cannot occur.