Orderability and Dehn filling

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 $π_{1} (M)$ into $\tilde{{PSL}_{2} ℝ}$ , which we organize into an infinite graph in $H^{1} (\partial M; ℝ)$ 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.

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Keywords

orderable groups, Dehn filling

Mathematical Subject Classification 2010

Primary: 57M60

Secondary: 57M25, 57M05, 20F60

References

Publication

Received: 12 February 2016

Revised: 2 March 2017

Accepted: 15 April 2017

Published: 16 March 2018

Proposed: Cameron Gordon

Seconded: András I. Stipsicz, Ciprian Manolescu

Authors

Marc Culler
Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago Chicago, IL United States
http://math.uic.edu/~culler
Nathan M Dunfield
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States
http://dunfield.info

Volume 22, issue 3 (2018)

Marc Culler and Nathan M Dunfield

Abstract