Left-orderable fundamental groups and Dehn surgery

TL;DR

This paper investigates how Dehn surgery affects the left-orderability of the fundamental group of 3-manifolds, providing criteria to determine when positive surgeries produce non-left-orderable groups, with applications to hyperbolic knots and L-spaces.

Contribution

It establishes a new criterion linking peripheral elements to left-orderability after Dehn surgery and applies it to infinite families of hyperbolic knots, including pretzel knots.

Findings

Sufficiently positive Dehn surgery can produce non-left-orderable fundamental groups.

Identifies conditions on peripheral elements that influence left-orderability post-surgery.

Examples include hyperbolic L-space knots where positive surgery yields L-spaces with non-left-orderable groups.

Abstract

There are various results that frame left-orderability of a group as a geometric property. Indeed, the fundamental group of a 3-manifold is left-orderable whenever the first Betti number is positive; in the case that the first Betti number is zero this property is closely tied to the existence of certain nice foliations. As a result, many large classes of 3-manifolds, including knot complements, are known to have left-orderable fundamental group. However, though the complement of a knot has left-orderable fundamental group, the result of Dehn surgery is a closed 3-manifold that need not have this property. We take this as motivation for the study of left-orderability in the context of Dehn surgery, and establish a condition on peripheral elements that must hold whenever a given Dehn surgery yields a manifold with left-orderable fundamental group. This leads to a workable criterion used to determine when sufficiently positive Dehn surgery produces manifolds with non-left-orderable fundamental group. As examples we produce infinite families of hyperbolic knots -- subsuming the (-2,3,q)-pretzel knots -- for which sufficiently positive surgery always produces a manifold with non-left-orderable fundamental group. Our examples are consistent with the observation that many (indeed, all known) examples of L-spaces have non-left-orderable fundamental group, as the given families of knots are hyperbolic L-space knots. Moreover, the behaviour of the examples studied here is consistent with the property that sufficiently positive surgery on an L-space knot always yields an L-space.