Universal circles for Anosov foliations

TL;DR

This paper investigates the universal circle constructions for Anosov foliations in 3-manifolds, revealing that different methods often produce distinct circles and that the universal circle from flow space boundary is generally not conjugate to others.

Contribution

It demonstrates the non-uniqueness of universal circles for Anosov foliations and introduces a flow space approach to parameterize the circle bundle at infinity.

Findings

Different universal circle constructions are typically distinct.

The universal circle from flow space boundary is not conjugate to others.

Flow space can be used to parameterize the circle bundle at infinity.

Abstract

Thurston introduced the notion of a universal circle associated to a taut foliation of a $3$-manifold as a way of organizing the ideal circle boundaries of its leaves into a single circle action. Calegari--Dunfield proved that every taut foliation of an atoroidal $3$-manifold $M$ has a universal circle, but the uniqueness (or lack-thereof) of this structure remains rather mysterious. In this paper, we consider the foliations associated to an Anosov flow $\varphi$ on $M$, showing that several constructions of a universal circle in the literature are typically distinct. Moreover, the underlying action of the Calegari--Dunfield leftmost universal circle is generally not even conjugate to the universal circle arising from the boundary of the flow space of $\varphi$. Our primary tool is a way to use the flow space of $\varphi$ to parameterize the circle bundle at infinity of $\varphi$'s invariant foliations.