Completing Prelaminations

TL;DR

This paper characterizes when pairs of transverse laminations on the circle can be extended to transverse foliations of the plane, with applications to flows on 3-manifolds and group actions.

Contribution

It provides a general framework for completing pairs of laminations to bifoliations, including singular cases, applicable to pseudo-Anosov flows and group actions.

Findings

Characterization of when lamination pairs can be completed to bifoliations.

Extension of the theory to singular bifoliations with prongs.

Applicability to flows on 3-manifolds and group actions.

Abstract

Motivated by problems in the study of Anosov and pseudo-Anosov flows on 3-manifolds, we characterize when a pair $(L^+, L^-)$ of subsets of transverse laminations of the circle can be completed to a pair of transverse foliations of the plane or, separately, realized as the endpoints of such a bifoliation of the plane. (We allow also singular bifoliations with simple prongs, such as arise in pseudo-Anosov flows). This program is carried out at a level of generality applicable to bifoliations coming from pseudo-Anosov flows with and without perfect fits, as well as many other examples, and is natural with respect to group actions preserving these structures.