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Generalized Torsion Elements and Bi-orderability of 3-manifold Groups

Published online by Cambridge University Press:  20 November 2018

Kimihiko Motegi  and
Masakazu Teragaito
Kimihiko Motegi
Affiliation:
Department of Mathematics, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156-8550, Japan e-mail: motegi@math.chs.nihon-u.ac.jp
Masakazu Teragaito
Affiliation:
Department of Mathematics and Mathematics Education, Hiroshima University, 1-1-1 Kagamiyama, Higashi-hiroshima 739-8524, Japan. e-mail: teragai@hiroshima-u.ac.jp
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Abstract

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It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group $F(2,m)\,(m>2)$ is a generalized torsion element.

Keywords

57M2557M0506F1520F05generalized torsion elementbi-orderingh-manifold group
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 4 , 01 December 2017 , pp. 830 - 844
Copyright
Copyright © Canadian Mathematical Society 2017

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