Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Generalized torsion and decomposition of $ 3$-manifolds


Authors: Tetsuya Ito, Kimihiko Motegi and Masakazu Teragaito
Journal: Proc. Amer. Math. Soc. 147 (2019), 4999-5008
MSC (2010): Primary 57M05, 20E06; Secondary 06F15, 20F60
DOI: https://doi.org/10.1090/proc/14581
Published electronically: July 9, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable $ 3$-manifold $ M$ has a generalized torsion element if and only if the fundamental group of some prime factor of $ M$ has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal $ 3$-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements. Additionally, in the course of the proof of the first result, we give an upper bound for the stable commutator length of generalized torsion elements.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57M05, 20E06, 06F15, 20F60

Retrieve articles in all journals with MSC (2010): 57M05, 20E06, 06F15, 20F60


Additional Information

Tetsuya Ito
Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
Email: tetitoh@math.kyoto-u.ac.jp

Kimihiko Motegi
Affiliation: Department of Mathematics, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156–8550, Japan
Email: motegi.kimihiko@nihon-u.ac.jp

Masakazu Teragaito
Affiliation: Department of Mathematics and Mathematics Education, Hiroshima University, 1-1-1 Kagamiyama, Higashi-Hiroshima, 739–8524, Japan
Email: teragai@hiroshima-u.ac.jp

DOI: https://doi.org/10.1090/proc/14581
Keywords: Fundamental group, generalized torsion element, stable commutator length, free product, free product with amalgamation, graph of groups, prime decomposition, torus decomposition
Received by editor(s): November 26, 2018
Received by editor(s) in revised form: February 5, 2019
Published electronically: July 9, 2019
Additional Notes: The first named author was partially supported by JSPS KAKENHI Grants Number JP15K17540 and JP16H02145.
The second named author was partially supported by JSPS KAKENHI Grant Number JP26400099 and Joint Research Grant of Institute of Natural Sciences at Nihon University for 2018.
The third named author was partially supported by JSPS KAKENHI Grant Number JP16K05149.
Communicated by: David Futer
Article copyright: © Copyright 2019 American Mathematical Society