Related papers: Generalized torsion for knots with arbitrarily hig…

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…

A generalized torsion in a group, an non-trivial element such that some products of its conjugates is the identity. This is an obstruction for a group being bi-orderable. Though it is known that there is a non bi-orderable group without…

In a group, a generalized torsion element is a non-identity element whose some non-empty finite product of its conjugates yields the identity. Such an element is an obstruction for a group to be bi-orderable. We show that the Weeks…

A non-trivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for…

It is well known that any knot group is torsion-free, but it may admit a generalized torsion element. We show that the knot group of any negative twist knot admits a generalized torsion element. This is a generalization of the same claim…

A group element is called a generalized torsion if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the…

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…

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 trivial element, then $g$ is called a generalized torsion element. To the best of our knowledge, we have no…

A group element is called generalized torsion if a finite product of its conjugates is equal to the identity. We show that in a finitely generated abelian-by-finite group, an element is generalized torsion if and only if its image in the…

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for…

A generalized torsion element is a non-trivial element such that some non-empty finite product of its conjugates is the identity. We construct a generalized torsion element of the fundamental group of a 3-manifold obtained by Dehn surgery…

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…

We show that a group whose generalized torsion elements are torsion elements (which we call a $TR^{*}$-group) is torsion-by-$R^{*}$ group, an extension of torsion group by a group without generalized torsion elements. We also discuss a…

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…

Let $G$ be a group. If an equation $x^n = y^n$ in $G$ implies $x = y$ for any elements $x$ and $y$, then $G$ is called an $R$--group. It is completely understood which knot groups are $R$--groups. Fay and Walls introduced $\bar{R}$--group…

The group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting number one contains infinitely many elements, none the automorphic image of another, such that each normally generates the group.

Given $\mathbf{n}=(n_{1},\ldots,n_{r})\in\mathbb{N}^r$, let $\Gamma_{\mathbf{n}}$ be a group presentable as $$\left\langle \gamma_{1},\ldots,\gamma_{r}\:|\:\gamma_{1}^{n_{1}}=\gamma_{2}^{n_{2}}=\cdots=\gamma_{r}^{n_{r}}\right\rangle. $$ If…

We give a condition sufficient to ensure that an amalgam of groups is generalized torsion-free. As applications, we construct a closed 3-manifold whose fundamental group is generalized torsion-free and non bi-orderable; a one-relator group…

We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the…

We classify knot traces with trisection genus at most 2. We give infinitely many knots whose traces have trisection genus 3, and infinitely many knots whose traces have trisection genus 4. We also show that there exist infinite families of…