Generalized Torsion in Knot Groups

Abstract

Abstract. In a group, a nonidentity element is called a generalized torsion element if some product of its conjugates equals the identity. We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot 5 2 and algebraic knots in the sense of Milnor.

Generalized torsions in once punctured torus bundles

“…However, it is conjectured that bi-orderability and having no generalized torsions is equivalent for the class of 3-manifold groups [8]. We have many affirmative answers for this conjecture, for some geometric manifolds, for some manifolds with non-trivial geometric decompositions, for many link complements and their Dehn fillings [5], [6], [8], [9], [11], [14], [15].…”

Generalized torsion for hyperbolic $3$--manifold groups with arbitrary large rank

“…Let us restrict our attention to hyperbolic 3-manifold groups. Two-generator, one-relator hyperbolic 3-manifold groups with generalized torsion elements are given in [10,14,15,8]. On the other hand, to the best of our knowledge, we have no hyperbolic 3-manifold groups with generalized torsion elements whose rank is explicitly known to be greater than two.…”

Generalized Torsion Elements and Bi-orderability of 3-manifold Groups

“…There are several works on the bi-orderability and generalized torsion elements of knot groups. The knot group of any torus knot is not bi-orderable, because it contains generalized torsion elements [23]. Thus Conjecture 1.1 holds for torus knot groups.…”