Floer homology, group orderability, and taut foliations of hyperbolic 3-manifolds

TL;DR

This paper investigates the conjectured equivalence among three properties of rational homology 3-spheres, providing extensive evidence through new methods and criteria applied to over 300,000 hyperbolic cases.

Contribution

It introduces a new combinatorial criterion called foliar orientation and applies it alongside other methods to study the properties for a large class of 3-manifolds.

Findings

Evidence supporting the conjecture for hyperbolic rational homology 3-spheres.

Development of a new combinatorial criterion for taut foliations.

Enhanced methods for analyzing Heegaard Floer homology and group orderability.

Abstract

This paper explores the conjecture that the following are equivalent for rational homology 3-spheres: having left-orderable fundamental group, having non-minimal Heegaard Floer homology, and admitting a co-orientable taut foliation. In particular, it adds further evidence in favor of this conjecture by studying these three properties for more than 300,000 hyperbolic rational homology 3-spheres. New or much improved methods for studying each of these properties form the bulk of the paper, including a new combinatorial criterion, called a foliar orientation, for showing that a 3-manifold has a taut foliation.

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