Classification of generalized torsion elements of order two in 3-manifold groups

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 identity, then g is called a generalized torsion element. The minimum number of conjugates in such a product is called the order of g. We will classify 3-manifolds M, each of whose fundamental group has a generalized torsion element of order two. Furthermore, we will classify such elements in pi 1(M). We also prove that R-group and R-group coincide for 3-manifold groups, and classify 3-manifold groups which are R-groups (and hence R-groups). (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.