Left-orderability in Dehn fillings of pseudo-Anosov mapping tori

TL;DR

This paper proves that all Dehn fillings of certain pseudo-Anosov mapping tori with specified properties have left-orderable fundamental groups, using two different foliation-based approaches, and verifies the L-space conjecture for specific knot surgeries.

Contribution

It introduces two novel foliation-based methods to establish left-orderability of fundamental groups in this context, and applies these results to confirm the L-space conjecture for certain pretzel knot surgeries.

Findings

All such Dehn fillings have left-orderable fundamental groups.

Two approaches produce explicit orders and representations related to the fillings.

Verification of the L-space conjecture for surgeries on the $(-2,3,2k+1)$-pretzel knot.

Abstract

For pseudo-Anosov mapping tori with co-orientable invariant foliations and monodromies reversing their co-orientations, a family of taut foliations was constructed in previous work on Dehn fillings with all rational slopes outside a neighborhood of the degeneracy slope. In this paper, we prove that all such Dehn fillings have left-orderable fundamental groups. We present two approaches, both based on an analysis of the branching behavior from such taut foliations. The first approach produces an $\mathbb{R}$-covered foliation arising from this family for each filling slope, and the second approach shows that, depending on the choice of a suitable system of arcs on $\Sigma$, one obtains a foliation that either has one-sided branching or is $\mathbb{R}$-covered. Consequently, the second approach associates to each Dehn filling a family of representations of its fundamental group into $\mathcal{G}_\infty$, the group of germs at infinity, whereas the first approach yields an explicit left-invariant order. As an application, combining our results with earlier work in the literature, we verify the L-space conjecture for all surgeries on the $(-2,3,2k+1)$-pretzel knot ($k \geqslant 3$) in $S^3$.