On laminar groups, Tits alternatives, and convergence group actions on \(S^2\)

Abstract

Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. A certain type of such groups---so-called pseudo-fibered groups---have previously been conjectured to be closely related to fundamental groups of closed hyperbolic 3-manifolds. We clarify this conjecture, and explain the necessity of various conditions. We then prove that torsion-free pseudo-fibered groups satisfy a Tits alternative. We conclude by proving that a purely hyperbolic pseudo-fibered group acts on the 2-sphere as a convergence group. This leads to a conjectural characterization of the fundamental group of a fibered hyperbolic 3-manifold as a pseudo-fibered group with some additional properties.