8,785 research outputs found
Some power of an element in a Garside group is conjugate to a periodically geodesic element
We show that for each element of a Garside group, there exists a positive
integer such that is conjugate to a periodically geodesic element
, an element with |h^n|_\D=|n|\cdot|h|_\D for all integers , where
|g|_\D denotes the shortest word length of 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 be a Garside group with Garside element , and let
be the minimal positive central power of . An element is said
to be 'periodic' if some power of it is a power of . 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 is cyclic; if
for some nonzero integer , then is conjugate to
; every finite subgroup of the quotient group is
cyclic.
By a classical theorem of Brouwer, Ker\'ekj\'art\'o and Eilenberg, an
-braid is periodic if and only if it is conjugate to a power of one of two
specific roots of . We generalize this to Garside groups by showing
that every periodic element is conjugate to a power of a root of .
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 , ,
, and the braid group of the complex reflection group of type
, 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
- β¦