On the centralizer of generic braids

We study the centralizer of a braid from the point of view of Garside theory, showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure. We present an algorithm to compute the centralizer of a braid whose generic-case complexity is quadratic on the length of the input, and which outputs a minimal set of generators in the generic case.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

On the centralizer of generic braids

We study the centralizer of a braid from the point of view of Garside theory, showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure. We present an algorithm to compute the centralizer of a braid whose generic-case complexity is quadratic on the length of the input, and which outputs a minimal set of generators in the generic case.Comment: 19 pages, 3 figure

Conjugacy in Garside groups I: Cyclings, powers, and rigidity

In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given X in a Garside group, if some power X^m is conjugate to a rigid element, then m can be bounded above by ||\Delta||^3. In the particular case of braid groups, this implies that a pseudo-Anosov braid has a small power whose ultra summit set consists of rigid elements. This solves one of the problems in the way of a polynomial solution to the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in braid groups. In addition to proving the rigidity theorem, it will be shown how this paper fits into the authors' program for finding a polynomial algorithm to the CDP/CSP, and what remains to be done.Comment: 41 page

A Garside-theoretic approach to the reducibility problem in braid groups

Let $D_n$ denote the $n$-punctured disk in the complex plane, where the punctures are on the real axis. An $n$-braid $\alpha$ is said to be \emph{reducible} if there exists an essential curve system \C in $D_n$, called a \emph{reduction system} of $\alpha$, such that \alpha*\C=\C where \alpha*\C denotes the action of the braid $\alpha$ on the curve system \C. A curve system \C in $D_n$ is said to be \emph{standard} if each of its components is isotopic to a round circle centered at the real axis. In this paper, we study the characteristics of the braids sending a curve system to a standard curve system, and then the characteristics of the conjugacy classes of reducible braids. For an essential curve system \C in $D_n$, we define the \emph{standardizer} of \C as \St(\C)=\{P\in B_n^+:P*\C{is standard}\} and show that \St(\C) is a sublattice of $B_n^+$. In particular, there exists a unique minimal element in \St(\C). Exploiting the minimal elements of standardizers together with canonical reduction systems of reducible braids, we define the outermost component of reducible braids, and then show that, for the reducible braids whose outermost component is simpler than the whole braid (including split braids), each element of its ultra summit set has a standard reduction system. This implies that, for such braids, finding a reduction system is as easy as finding a single element of the ultra summit set.Comment: 38 pages, 18 figures, published versio

Abelian subgroups of Garside groups

In this paper, we show that for every abelian subgroup $H$ of a Garside group, some conjugate $g^{-1}Hg$ consists of ultra summit elements and the centralizer of $H$ is a finite index subgroup of the normalizer of $H$. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the algebraic flat torus theorem for Garside groups and solve several algorithmic problems concerning abelian subgroups of Garside groups.Comment: This article replaces our earlier preprint "Stable super summit sets in Garside groups", arXiv:math.GT/060258

On the cycling operation in braid groups

The cycling operation is a special kind of conjugation that can be applied to elements in Artin's braid groups, in order to reduce their length. It is a key ingredient of the usual solutions to the conjugacy problem in braid groups. In their seminal paper on braid-cryptography, Ko, Lee et al. proposed the {\it cycling problem} as a hard problem in braid groups that could be interesting for cryptography. In this paper we give a polynomial solution to that problem, mainly by showing that cycling is surjective, and using a result by Maffre which shows that pre-images under cycling can be computed fast. This result also holds in every Artin-Tits group of spherical type. On the other hand, the conjugacy search problem in braid groups is usually solved by computing some finite sets called (left) ultra summit sets (left-USS), using left normal forms of braids. But one can equally use right normal forms and compute right-USS's. Hard instances of the conjugacy search problem correspond to elements having big (left and right) USS's. One may think that even if some element has a big left-USS, it could possibly have a small right-USS. We show that this is not the case in the important particular case of rigid braids. More precisely, we show that the left-USS and the right-USS of a given rigid braid determine isomorphic graphs, with the arrows reversed, the isomorphism being defined using iterated cycling. We conjecture that the same is true for every element, not necessarily rigid, in braid groups and Artin-Tits groups of spherical type.Comment: 20 page