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Non-Left-Orderable 3-Manifold Groups

Published:2005-03-01
Printed: Mar 2005
• Mieczysław K. Dąbkowski
• Józef H. Przytycki
• Amir A. Togha
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Abstract

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of $S^3$ 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 $S^3$ branched along various hyperbolic 2-bridge knots. %with various hyperbolic 2-bridge knots as branched sets. 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.
 MSC Classifications: 57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M12 - Special coverings, e.g. branched 20F60 - Ordered groups [See mainly 06F15]