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Complicated generalized torsion elements in Seifert fibered spaces with boundary

Graduate School of Advanced Science and Engineering, Hiroshima University, 1-3-1 Kagamiyama, Higashi-hiroshima 7398526, Japan

E-mail Address: himeno-keisuke@hiroshima-u.ac.jp

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https://doi.org/10.1142/S0218216523500803Cited by:0 (Source: Crossref)

Abstract

In a group, a nontrivial element is called a generalized torsion element if some non-empty finite product of its conjugates is equal to the identity. There are various examples of torsion-free groups which contain generalized torsion elements. We can define the order of a generalized torsion element as the minimum number of its conjugates required to generate the identity. In previous works, three-manifold groups which contain a generalized torsion element of order two are determined. However, there are few previous studies that examine the order of a generalized torsion element bigger than two. In this paper, we focus on Seifert fibered spaces with boundary, including the torus knot exteriors, and construct concretely generalized torsion elements of order $3$ , $4$ , $6$ and others in their fundamental groups.

Keywords:

AMSC: 57K10, 06F15, 20F60

References

1. M. Aschenbrenner, S. Friedl and H. Wilton , 3-Manifold Groups, EMS Series of Lectures in Mathmatics, Vol. 20 (European Mathmatical Society, Zurich, 2015), xiv+215 pp. Crossref, Google Scholar
2. C. Bavard , Longueur stable des commutateurs, Enseign. Math. 37(1–2) (1991) 109–150. Google Scholar
3. S. Boyer, D. Rolfsen and W. Wiest , Orderable $3$ -manifold groups, Ann. Inst. Fourier 55 (2005) 243–288. Crossref, Web of Science, Google Scholar
4. D. Calegari , Scl, MSJ Memoirs, Mathematical Society of Japan, Vol. 20 (The Mathematical Society of Japan, Tokyo, 2009), xii+209 pp. Crossref, Google Scholar
5. L. Chen , Spectral gap of scl in free products, Proc. Amer. Math. Soc. 146(7) (2018) 3143–3151. Crossref, Web of Science, Google Scholar
6. L. Chen , Scl in free products, Algebra Geom. Topol. 18(6) (2018) 3279–3313. Crossref, Web of Science, Google Scholar
7. A. Clay and D. Rolfsen , Ordered Groups and Topology, Graduate Studies in Mathematics, Vol. 176 (American Mathematical Society, Providence, RI, 2016), x+154 pp. Crossref, Google Scholar
8. A. Hatcher, Notes on basic 3-manifold topology, available from the author’s website (2007), https://pi.math.cornell.edu/hatcher/3M/3Mfds.pdf. Google Scholar
9. K. Himeno, K. Motegi and M. Teragaito , Generalized torsion, unique root property and Baumslag–Solitar relation for knot groups, Hiroshima Math. J. 53 (2023) 1–14. Crossref, Web of Science, Google Scholar
10. K. Himeno, K. Motegi and M. Teragaito, Classification of generalized torsion elements of order two in $3$ -manifold groups, preprint. Google Scholar
11. T. Ito, K. Motegi and M. Teragaito , Generalized torsion and decomposition of $3$ -manifolds, Proc. Amer. Math. Soc. 147(11) (2019) 4999–5008. Crossref, Web of Science, Google Scholar
12. W. Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, Vol. 43 (American Mathematical Society, Providence, R.I., 1980), xii+251 pp. Google Scholar
13. K. Motegi and M. Teragaito , Generalized torsion elements and bi-orderability of 3-manifold groups, Canad. Math. Bull. 60(4) (2017) 830–844. Crossref, Web of Science, Google Scholar
14. W. Magnus, A. Karrass and D. Solitar , Combinatorial Group Theory. Presentations of Groups in Terms of Generators and Relations (Dover Publications, Inc., Mineola, New York, 2004), xii+444 pp. Google Scholar
15. G. Naylor and D. Rolfsen , Generalized torsion in knot groups, Canad. Math. Bull. 59(1) (2016) 182–189. Crossref, Web of Science, Google Scholar
16. A. Walker , Stable commutator length in free products of cyclic groups, Exp. Math. 22(3) (2013) 282–298. Crossref, Web of Science, Google Scholar
17. A. Walker, Scallop, a computer program, https://github.com/aldenwalker/scallop. Google Scholar