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Non-Left-Orderable 3-Manifold Groups

Published online by Cambridge University Press:  20 November 2018

Mieczysław K. Dąbkowski,
Józef H. Przytycki and
Amir A. Togha
Mieczysław K. Dąbkowski
Affiliation:
Mathematical Science Department, The Univesity of Texas at Dallas, Richardson, TX 75803-0688, U.S.A. e-mail: mdab@utdallas.edu
Józef H. Przytycki
Affiliation:
Department of Mathematics, The GeorgeWashington University, Washington, D.C. 20052, U.S.A. e-mail: przytyck@gwu.edu
Amir A. Togha
Affiliation:
Department of Mathematics, The GeorgeWashington University, Washington, D.C. 20052, U.S.A. e-mail: userid@gwu.edu
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Abstract

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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. 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.

Keywords

57M2557M1220F60

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 1 , 01 March 2005, pp. 32 - 40
Copyright
Copyright © Canadian Mathematical Society 2005

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