#### Volume 22, issue 3 (2018)

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Orderability and Dehn filling

### Marc Culler and Nathan M Dunfield

Geometry & Topology 22 (2018) 1405–1457
##### 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 ${\pi }_{1}\left(M\right)$ into $\stackrel{˜}{{PSL}_{2}ℝ}$, which we organize into an infinite graph in ${H}^{1}\left(\partial M;ℝ\right)$ 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.

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##### Keywords
orderable groups, Dehn filling
##### Mathematical Subject Classification 2010
Primary: 57M60
Secondary: 57M25, 57M05, 20F60