Generalized torsion elements in the fundamental groups of 3-manifolds obtained by the 0-surgeries along some double twist knots

Abstract

We consider the 3-manifold obtained by the 0-surgery along a double twist knot. We construct a candidate for a generalized torsion element in the fundamental group of the surged manifold, and see that there exists the cases where the candidate is actually a generalized torsion element. For a proof, we use the JSJ-decomposition of the surged manifold. We also prove that the fundamental group of the 3-manifold obtained from the 0-surgery along a double twist knot is bi-orderable if and only if it admits no generalized torsion elements. We also list some examples of the surged manifolds whose fundamental groups admit generalized torsion elements.