Groups acting on veering pairs and Kleinian groups

TL;DR

This paper demonstrates that laminar groups with invariant veering pairs are hyperbolic 3-orbifold groups, constructing loom spaces from veering pairs without assuming initial 3-manifold structures, thus advancing geometrization theory.

Contribution

It establishes a new link between veering pairs of laminations and hyperbolic 3-orbifold groups, introducing a method to construct loom spaces without initial manifold assumptions.

Findings

Laminar groups with invariant veering pairs are hyperbolic 3-orbifold groups

Constructs loom spaces from veering pairs without initial 3-manifold assumptions

Provides a geometrization-type result surpassing previous relations among veering triangulations and flows

Abstract

We show that some laminar group which has an invariant veering pair of laminations is a hyperbolic 3-orbifold group. On the way, we show that from a veering pair of laminations, one can construct a loom space (in the sense of Schleimer-Segerman) as a quotient. Our approach does not assume the existence of any 3-manifold to begin with so this is a geometrization-type result, and supersedes some of the results regarding the relation among veering triangulations, pseudo-Anosov flows, taut foliations in the literature.