Simultaneous universal circles

TL;DR

This paper constructs a universal circle for taut foliations almost transverse to pseudo-Anosov flows on atoroidal 3-manifolds, linking the flow space boundary with the foliation's structure.

Contribution

It introduces a method to define a universal circle for such foliations using the flow space boundary and explicit monotone maps, extending Thurston and Calegari-Dunfield's framework.

Findings

Universal circle constructed for taut foliations almost transverse to pseudo-Anosov flows.

Defines explicit monotone maps linking flow space boundary to foliation structure.

Provides a new tool for understanding foliation dynamics in 3-manifolds.

Abstract

Let phi be a pseudo-Anosov flow on a closed oriented atoroidal 3-manifold M. We show that if F is any taut foliation almost transverse to phi, then the action of pi_1(M) on the boundary of the flow space, together with a natural collection of explicitly described monotone maps, defines a universal circle for F in the sense of Thurston and Calegari-Dunfield.