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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
Braid groups of imprimitive complex reflection groups
We obtain new presentations for the imprimitive complex reflection groups of
type and their braid groups for . Diagrams
for these presentations are proposed. The presentations have much in common
with Coxeter presentations of real reflection groups. They are positive and
homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms
correspond to group automorphisms. The new presentation shows how the braid
group is a semidirect product of the braid group of affine type
and an infinite cyclic group. Elements of are
visualized as geometric braids on strings whose first string is pure and
whose winding number is a multiple of . We classify periodic elements, and
show that the roots are unique up to conjugacy and that the braid group
is strongly translation discrete.Comment: published versio
An upper bound of the minimal asymptotic translation length of right-angled Artin groups on extension graphs
For the right-angled Artin group action on the extension graph, it is known
that the minimal asymptotic translation length is bounded above by 2 provided
that the defining graph has diameter at least 3. In this paper, we show that
the same result holds without any assumption. This is done by exploring some
graph theoretic properties of biconnected graphs, i.e. connected graphs whose
complement is also connected
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