Some power of an element in a Garside group is conjugate to a periodically geodesic element

We show that for each element $g$ of a Garside group, there exists a positive integer $m$ such that $g^m$ is conjugate to a periodically geodesic element $h$, an element with |h^n|_\D=|n|\cdot|h|_\D for all integers $n$, where |g|_\D denotes the shortest word length of $g$ with respect to the set \D of simple elements. We also show that there is a finite-time algorithm that computes, given an element of a Garside group, its stable super summit set.Comment: Subj-class of this paper should be Geometric Topology; Version published by BLM

Periodic elements in Garside groups

Let $G$ be a Garside group with Garside element $\Delta$, and let $\Delta^m$ be the minimal positive central power of $\Delta$. An element $g\in G$ is said to be 'periodic' if some power of it is a power of $\Delta$. In this paper, we study periodic elements in Garside groups and their conjugacy classes. We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of $G$ is cyclic; if $g^k=\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\Delta^a$; every finite subgroup of the quotient group $G/$ is cyclic. By a classical theorem of Brouwer, Ker\'ekj\'art\'o and Eilenberg, an $n$-braid is periodic if and only if it is conjugate to a power of one of two specific roots of $\Delta^2$. We generalize this to Garside groups by showing that every periodic element is conjugate to a power of a root of $\Delta^m$. We introduce the notions of slimness and precentrality for periodic elements, and show that the super summit set of a slim, precentral periodic element is closed under any partial cycling. For the conjugacy problem, we may assume the slimness without loss of generality. For the Artin groups of type $A_n$, $B_n$, $D_n$, $I_2(e)$ and the braid group of the complex reflection group of type $(e,e,n)$, endowed with the dual Garside structure, we may further assume the precentrality.Comment: The contents of the 8-page paper "Notes on periodic elements of Garside groups" (arXiv:0808.0308) have been subsumed into this version. 27 page

Spin Hall Effect-driven Spin Torque in Magnetic Texture

Current-induced spin torque and magnetization dynamics in the presence of spin Hall effect in magnetic textures is studied theoretically. The local deviation of the charge current gives rise to a current-induced spin torque of the form (1-\beta{\bf M})x[({\bf u}_0+\alpha_H{\bf u}_0x\bf M})\cdot{\bm \nabla}]{\bf M}, where {\bf u}_0 is the direction of the injected current, \alpha_H is the Hall angle and \beta is the non-adiabaticity parameter due to spin relaxation. Since \alpha_H and \beta can have a comparable order of magnitude, we show that this torque can significantly modify the current-induced dynamics of both transverse and vortex walls.Comment: 3 pages, 1 Figur