Generalized torsion for hyperbolic 3-manifold groups with arbitrary large rank. (English) Zbl 1522.57013
Summary: Let \(G\) be a group and \(g\) a non-trivial element in \(G\). If some non-empty finite product of conjugates of \(g\) equals to the trivial element, then \(g\) is called a generalized torsion element. To the best of our knowledge, we have no hyperbolic 3-manifold groups with generalized torsion elements whose rank is explicitly known to be greater than two. The aim of this short note is to demonstrate that for a given integer \(n > 1\) there are infinitely many closed hyperbolic 3-manifolds \(M_n\) which enjoy the property: (i) the Heegaard genus of \(M_n\) is \(n\), (ii) the rank of \(\pi_1(M_n)\) is \(n\), and (ii) \(\pi_1(M_n)\) has a generalized torsion element. Furthermore, we may choose \(M_n\) as homology lens spaces and so that the order of the generalized torsion element is arbitrarily large.
{© 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}
{© 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}
MSC:
| 57K10 | Knot theory |
| 57M05 | Fundamental group, presentations, free differential calculus |
| 57M07 | Topological methods in group theory |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 06F15 | Ordered groups |
| 20F05 | Generators, relations, and presentations of groups |
| 20F60 | Ordered groups (group-theoretic aspects) |
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