Characterization of Fuchsian groups as laminar groups & the structure theorem of hyperbolic 2-orbifolds

TL;DR

This paper characterizes Fuchsian groups via invariant circle laminations and establishes a structure theorem for hyperbolic 2-orbifolds, generalizing previous results and providing new insights into their geometric and dynamical properties.

Contribution

It proves that groups with three transverse invariant circle laminations are Fuchsian, extending prior results to include orbifolds of infinite type and non-discrete groups.

Findings

Groups with three invariant circle laminations are Fuchsian.

A structure theorem for hyperbolic 2-orbifolds, including infinite type.

Complete generalization of previous results on circle actions.

Abstract

Thurston and Calegari-Dunfield showed that the fundamental group of some tautly foliated hyperbolic 3-manifold acts on the circle in a distinctive way that the action preserves some structure of S^1, so-called a circle lamination. Indeed, a large class of Kleinian groups acts on the circle preserving a circle lamination. In this paper, we are concerned with the converse problem that a group acting on the circle with at least two invariant circle laminations is Kleinian. We prove that a subgroup of the orientation preserving circle homeomorphism group is a Fuchsian group whose quotient orbifold is not a geometric pair of pants (or turnover) if and only if it preserves three circle laminations with a certain transversality. This is the complete generalization of the previous result of the first author which is proven under the assumption that the subgroup is discrete and torsion-free. On the way to our main result, we also show a structure theorem (generalized pants decomposition) for complete 2-dimensional hyperbolic orbifolds, including the case of infinite type, which is of independent interest.