Depth-one foliations, pseudo-Anosov flows and universal circles

TL;DR

This paper establishes a deep connection between universal circles from depth-one foliations and the ideal boundary of flow spaces in 3-manifolds, revealing uniqueness of certain pseudo-Anosov flows.

Contribution

It proves the isomorphism between the universal circle and the flow space boundary, and shows the uniqueness of pseudo-Anosov flows without perfect fits transverse to the foliation.

Findings

Universal circle is isomorphic to the flow space boundary.

At most one pseudo-Anosov flow without perfect fits exists up to orbit equivalence.

The result applies to closed atoroidal 3-manifolds with depth-one foliations.

Abstract

Given a taut depth-one foliation $\mathcal{F}$ in a closed atoroidal 3-manifold $M$ transverse to a pseudo-Anosov flow $\phi$ without perfect fits, we show that the universal circle coming from leftmost sections $\mathfrak{S}_\mathrm{left}$ associated to $\mathcal{F}$, constructed by Thurston and Calegari-Dunfield, is isomorphic to the ideal boundary of the flow space associated to $\phi$ with natural structure maps. As a corollary, we use a theorem of Barthelm\'e-Frankel-Mann to show that there is at most one pseudo-Anosov flow without perfect fits transverse to $\mathcal{F}$ up to orbit equivalence.