Orderability of homology spheres obtained by Dehn filling

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 $\widetilde{\mathrm{PSL}_2 \mathbb{R}}$ 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.

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Received 6 October 2019

Received revised 22 November 2021

Accepted 21 December 2021

Published 21 April 2023