14,615 research outputs found
The infimum, supremum and geodesic length of a braid conjugacy class
Algorithmic solutions to the conjugacy problem in the braid groups B_n were
given by Elrifai-Morton in 1994 and by the authors in 1998. Both solutions
yield two conjugacy class invariants which are known as `inf' and `sup'. A
problem which was left unsolved in both papers was the number m of times one
must `cycle' (resp. `decycle') in order to increase inf (resp. decrease sup) or
to be sure that it is already maximal (resp. minimal) for the given conjugacy
class. Our main result is to prove that m is bounded above by n-2 in the
situation of the second algorithm and by ((n^2-n)/2)-1 in the situation of the
first. As a corollary, we show that the computation of inf and sup is
polynomial in both word length and braid index, in both algorithms. The
integers inf and sup determine (but are not determined by) the shortest
geodesic length for elements in a conjugacy class, as defined by Charney, and
so we also obtain a polynomial-time algorithm for computing this geodesic
length.Comment: 15 pages. Journa
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
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
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