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 π₁(M). We also prove that R-group and R̄-group coincide for $3$-manifold groups, and classify $3$-manifold groups which are R-groups (and hence R̄-groups).