Orderability and Dehn filling

TL;DR

This paper develops a new technique using representations into the universal cover of PSL(2,R) to identify intervals of Dehn fillings with left-orderable fundamental groups, linking topology and group orderability.

Contribution

It introduces the translation extension locus as a tool to determine left-orderability of fundamental groups after Dehn filling, broadening understanding of 3-manifold group orderability.

Findings

Intervals of Dehn fillings with left-orderable groups identified

Translation extension locus visualizations support theoretical results

New connections between Floer homology, taut foliations, and orderability

Abstract

Motivated by conjectures relating group orderability, Floer homology, and taut foliations, we discuss a systematic and broadly applicable technique for constructing left-orders on the fundamental groups of rational homology 3-spheres. Specifically, for a compact 3-manifold $M$ with torus boundary, we give several criteria which imply that whole intervals of Dehn fillings of $M$ have left-orderable fundamental groups. Our technique uses certain representations from $\pi_1(M)$ into $\widetilde{\mathrm{PSL}_2 \mathbb{R}}$, which we organize into an infinite graph in $H^1(\partial M; \mathbb{R})$ called the translation extension locus. We include many plots of such loci which inform the proofs of our main results and suggest interesting avenues for future research.