Taut foliations, left-orders, and pseudo-Anosov mapping tori

TL;DR

This paper constructs actions of 3-manifold fundamental groups on the real line to study their transverse geometry, linking taut foliations, left-orderability, and hyperbolic manifold properties.

Contribution

It introduces a new method to analyze the transverse geometry of taut foliations via group actions, connecting to left-orderability and extending to numerous hyperbolic 3-manifolds.

Findings

Fundamental groups of non-trivial surgeries on the figure eight knot are left-orderable.

The techniques apply to at least 2598 hyperbolic 3-manifolds, covering 44.7% of certain rational homology spheres.

Constructed actions capture the transverse geometry of taut foliations, complementing Thurston's universal circle.

Abstract

For a large class of 3-manifolds with taut foliations, we construct an action of $\pi_1(M)$ on $\mathbb{R}$ by orientation preserving homeomorphisms which captures the transverse geometry of the leaves. This action is complementary to Thurston's universal circle. Applications include the left-orderability of the fundamental groups of every non-trivial surgery on the figure eight knot. Our techniques also apply to at least 2598 manifolds representing 44.7% of the non-L-space rational homology spheres in the Hodgson-Weeks census of small closed hyperbolic 3-manifolds.