Orderability of Homology Spheres Obtained by Dehn Filling

TL;DR

This paper introduces a method to determine when the fundamental groups of certain 3-manifolds obtained by Dehn filling are left-orderable, using holonomy extension loci to analyze representations.

Contribution

It develops the holonomy extension locus technique to identify intervals of Dehn fillings producing rational homology 3-spheres with left-orderable fundamental groups.

Findings

Constructed holonomy extension loci for rational homology solid tori.

Identified intervals of Dehn fillings with left-orderable fundamental groups.

Connected holonomy loci to left-orderability in rational homology 3-spheres.

Abstract

In this paper, we develop a method for constructing left-orders on the fundamental groups of rational homology 3-spheres. We begin by constructing the holonomy extension locus of a rational homology solid torus $M$, which encodes the information about peripherally hyperbolic $\widetilde{\text{PSL}_2\mathbb{R}}$ representations of $\pi_1(M)$. Plots of the holonomy extension loci of many rational homology solid tori are shown, and the relation to left-orderability is hinted. Using holonomy extension loci, we study rational homology 3-spheres coming from Dehn filling on rational homology solid tori and construct intervals of Dehn fillings with left-orderable fundamental group.