Generalized torsion and decomposition of 3-manifolds

Tetsuya Ito; Kimihiko Motegi; Masakazu Teragaito

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arXiv · 1811.07532

Generalized torsion and decomposition of 3-manifolds

Tetsuya Ito·Kimihiko Motegi·Masakazu Teragaito

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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.

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Tetsuya Ito, Kimihiko Motegi, Masakazu Teragaito. 2018-11-19. Generalized torsion and decomposition of 3-manifolds. https://arxiv.org/abs/1811.07532

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