On laminar groups, Tits alternatives, and convergence group actions on $S^2$

TL;DR

This paper explores properties of pseudo-fibered groups, a class of groups acting on the circle and 2-sphere, establishing their Tits alternative and convergence group actions, with implications for 3-manifold groups.

Contribution

It introduces and analyzes pseudo-fibered groups, proving they satisfy the Tits alternative and act as convergence groups on the 2-sphere, expanding understanding of their structure.

Findings

Many 3-manifold groups are pseudo-fibered.

Torsion-free pseudo-fibered groups satisfy the Tits alternative.

Hyperbolic pseudo-fibered groups act as convergence groups on $S^2$.

Abstract

Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups-so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups. 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 an interesting question if there are examples of pseudo-fibered groups other than 3-manifold groups.