Non-Left-Orderable 3-Manifold Groups

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of S³ branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of S³ branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the 5₂ knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.