Quasigeodesic flows and M\"obius-like groups

TL;DR

This paper investigates quasigeodesic flows on hyperbolic 3-manifolds, showing their fundamental groups act on a circle and exploring the nature of these actions, with implications for the existence of closed orbits.

Contribution

It establishes a natural circle action for fundamental groups of hyperbolic 3-manifolds with quasigeodesic flows and analyzes the M"obius-like nature of these actions.

Findings

Fundamental group acts on a circle by homeomorphisms.

Flow either has a closed orbit or boundary action is M"obius-like.

Conjecture that M"obius-like actions are not conjugate into PSL(2, R).

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