2018/11/19 by Tetsuya Ito, Ito, Tetsuya, Kimihiko Motegi +3
Mathematics · #20E06 (Primary)m 06F15 #20F60 (Secondary) #57M05 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1811.07532
openalex publication_date 2018/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.