Non-left-orderable 3-manifold groups

Abstract

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 5 2 knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume. ©Canadian Mathematical Society 2005.