ar X iv : m at h / 03 02 09 8 v 2 [ m at h . G T ] 1 1 M ay 2 00 4 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 S 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 S 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. We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. We show that several torsion free 3-manifold groups are not left-orderable. Many of our manifolds are obtained by taking n-fold branched covers along various hyperbolic 2-bridge knots. The paper is organized in the following way: after defining left-orderability we state our main theorem listing branched set links and multiplicity of coverings from which we obtain manifolds with non-left-orderable groups. Then we describe presentations of these groups in a way which allows the proof of non-left-orderability in a uniform way. The Main Lemma (Lemma 5) is the algebraic underpinning of our method and the non-left-orderability follows easily from it in almost all cases. Moreover we prove the non-left-orderability of a family of 3-manifold groups to which the Main Lemma does not apply. These groups, known as generalized Fibonacci groups F (n− 1, n), arise as groups of double covers of S branched along pretzel links of type (2, 2, ..., 2,−1). We end the paper with some questions and speculations. Definition 1 A group is left-orderable if there is a strict total ordering ≺ of its elements which is left-invariant: x ≺ y iff zx ≺ zy for all x, y and z. Straight from the definition, it follows that a group with a torsion element is not left-orderable.