Generalized torsion elements and hyperbolic links

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 manifold, the figure-eight sister manifold, and the complement of Whitehead sister link admit generalized torsion elements in their fundamental groups. In particular, the Whitehead sister link, which is the pretzel link of type [Formula: see text], can be generalized to hyperbolic pretzel links of type [Formula: see text]. These give the first examples of hyperbolic links whose link groups admit generalized torsion elements.

Torsion elements of order 𝑝2 in the Nottingham group

Let κ be a characteristic p finite field of q elements and {\mathfrak{N}_{\kappa}} the Nottingham group over κ. Lubin associated to every conjugacy class of torsion element of {\mathfrak{N}_{\kappa}} a type. We establish an upper bound {B(q;l,m)} on the number of conjugacy classes of order {p^{2}} torsion elements u of {\mathfrak{N}_{\kappa}} of type {\langle l,m\rangle}. In the case where {l<p}, the bound {B(q;l,m)} is the exact number of conjugacy classes. Moreover, we give a criterion on when u and {u^{n}} are conjugate.

Enhancing Mixing and Thermal Management of Recycled Carbon Composite Systems by Torsion-Induced Phase-to-Phase Thermal and Molecular Mobility

A novel torsion screw has been proposed to resolve the inadequate control of mass transfer and the thermal management of two component polymer blends and their carbon fiber composites. The novel torsional screw distinctly introduced radial flow in the torsion screw channel, which is a significant improvement over the flow pattern developed by the conventional screw. The heat transfer and mixing behavior of melt mixtures are enhanced by adapting screws with torsion elements compared with the traditional screw elements. Heat transfer efficacy in the polypropylene–polystyrene bi-phasic extrusion process improved with the increase in torsion element numbers. An increased number of newly designed torsional elements also improved the dispersion of minor phase in bi-phase polypropylene–polystyrene composition and their carbon fiber composites. The unique flow pattern induced by the torsion elements shows a synergistic effect on the melt-phase mass flow and the thermal flow field facilitating phase-to-phase thermal and molecular mobility and enhanced fiber orientation, crystallinity and mechanical properties of composite made from recycled carbon fiber/polypropylene. Microtomographs of recycled carbon fiber demonstrated the extraordinary ability of a torsion screw element to orient carbon fiber in both axial and radial directions.

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

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 non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group F(2,m) (m > 2) is a generalized torsion element.

LOCAL BORCHERDS PRODUCTS FOR UNITARY GROUPS

For the modular variety attached to an arithmetic subgroup of an indefinite unitary group of signature $(1,n+1)$, with $n\geqslant 1$, we study Heegner divisors in the local Picard group over a boundary component of a compactification. For this purpose, we introduce local Borcherds products. We obtain a precise criterion for local Heegner divisors to be torsion elements in the Picard group, and further, as an application, we show that the obstructions to a local Heegner divisor being a torsion element can be described by certain spaces of vector-valued elliptic cusp forms, transforming under a Weil representation.

Generalized Torsion in Knot Groups

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 ûnd 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 52, and algebraic knots in the sense of Milnor.

Conjugacy classes of torsion in GL_n(Z)

The problem of integral similarity of block-triangular matrices over the ring of integers is connected to that of finding representatives of the classes of an equivalence relation on general integer matrices. A complete list of representatives of conjugacy classes of torsion in the 4 × 4 general linear group over ring of integers is given. There are 45 distinct such classes and each torsion element has order of 1, 2, 3, 4, 5, 6, 8, 10 or 12.

On Homotopy Invariants of Combings of Three-manifolds

Combings of compact, oriented, 3-dimensional manifoldsMare homotopy classes of nowhere vanishing vector fields. The Euler class of the normal bundle is an invariant of the combing, and it only depends on the underlying Spinc-structure. A combing is called torsion if this Euler class is a torsion element of H2(M; Z). Gompf introduced a Q-valued invariant θGof torsion combings on closed 3-manifolds, and he showed that θGdistinguishes all torsion combings with the same Spinc-structure. We give an alternative definition for θGand we express its variation as a linking number. We define a similar invariantp1of combings for manifolds bounded by S2. We relate p1 to the Θ-invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formula Θ = ¼P1+ 6λ, where λ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.

Twisted gamma filtration and algebras with orthogonal involution

For the Grothendieck group of a split simple linear algebraic group, the twisted γ-filtration provides a useful tool for constructing torsion elements in -rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the γ-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline.