  This paper explores the conjecture that the following are equivalent for
rational homology 3-spheres: having left-orderable fundamental group, having
non-minimal Heegaard Floer homology, and admitting a co-orientable taut
foliation. In particular, it adds further evidence in favor of this conjecture
by studying these three properties for more than 300,000 hyperbolic rational
homology 3-spheres. New or much improved methods for studying each of these
properties form the bulk of the paper, including a new combinatorial criterion,
called a foliar orientation, for showing that a 3-manifold has a taut
foliation.
