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
–spheres. Specifically,
for a compact
–manifold
with
torus boundary, we give several criteria which imply that whole intervals of Dehn fillings
of
have
left-orderable fundamental groups. Our technique uses certain representations from
into
, which we organize into
an infinite graph in
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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