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.
- Tetsuya Ito
- Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
- MR Author ID: 922393
- ORCID: 0000-0001-8156-1341
- Email: firstname.lastname@example.org
- Kimihiko Motegi
- Affiliation: Department of Mathematics, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156–8550, Japan
- MR Author ID: 254668
- Email: email@example.com
- Masakazu Teragaito
- Affiliation: Department of Mathematics and Mathematics Education, Hiroshima University, 1-1-1 Kagamiyama, Higashi-Hiroshima, 739–8524, Japan
- MR Author ID: 264744
- Email: firstname.lastname@example.org
- 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
- © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 4011531