稳定的超级首脑会议集在Garside集团
Stable super summit sets in Garside groups
摘要：
撤回，并列入我们的新手稿“加塞德集团的阿贝尔子群”，数学。GT/0609683
英文摘要：
withdrawn and included in our new manuscript "Abelian subgroups of Garside groups", math.GT/0609683
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Garside群的共轭性II：超顶点集的结构
Conjugacy in Garside groups II: Structure of the ultra summit set
本文是研究Garside群上的共轭判定问题和共轭搜索问题的系列文章的第二篇。Garside群G中元素X的超顶点集USS（X）是G中元素的有限集，它是G中X的共轭类的完全不变量。如果人们希望找到USS（X）大小的界限，一个根本问题就是了解它的结构。本文引入USS（X）中元素的两种新的运算，称为“部分循环”和“部分扭去循环”，并证明了：如果Y和Z属于USS（X），则Y和Z由部分循环和部分扭去循环序列相关联.这些运算是理解由第二作者的论文中的凸性定理所得到的最小简单元的一种具体方法。利用部分循环和部分扭去循环，研究了由USS（X）确定的有向图Γ X的结构，证明了Γ X可分解为“黑”和“灰”子图.在本系列的第一篇文章中，有一些与该程序相关的应用程序，用于在编织物的情况下找到CDP/CSP的多项式解。一个不同的应用是给出一个新的算法来解决CDP/CSP在Garside组这是比所有其他已知的算法，即使其理论复杂性是相同的，由第二作者。也有应用到理论的约化群。
math.GRmath.GT
Conjugacy classes of periodic braids
Recently, there have been several progresses for the conjugacy search problem (CSP) in Garside groups, especially in braid groups. All known algorithms for solving this problem use a sort of exhaustive search in a particular finite set such as the super summit set and the ultra summit set. Their complexities are proportional to the size of the finite set, even when there exist very short conjugating elements. However, ultra summit sets are very large in some cases especially for reducible braids and periodic braids. Some possible approaches to resolve this difficulty would be either to use dif…
math.GRmath.GT
Small braids having a big Ultra Summit Set
In [J.Birman, V.Gebhardt, J.Gonzalez-Meneses, Conjugacy in Garside groups I: cyclings, powers and rigidity] authors asked: (open question 2) is the size of USS of a rigid pseudo-Anosov braid is bounded above by some polynomial in the number of strands and the braid length? We answer this question in the negative.
math.GRmath.GT
Garside群的Abel子群
Abelian subgroups of Garside groups
本文证明了对Garside群的任意交换子群$H$，某共轭$g^{-1}Hg$由超顶点元组成，且$H$的中心化子是$H$的正规化子的有限指数子群.结合Garside群中关于平移数的结果，得到了Garside群的代数平坦环面定理的一个简单证明，并解决了Garside群的交换子群的几个算法问题.
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Garside群中的循环滑动运算
The cyclic sliding operation in Garside groups
我们提出了一个新的操作要执行的元素在Garside组，称为循环滑动，这是介绍，以取代众所周知的循环和decycling操作。循环滑动似乎是一个更自然的选择，简化了Garside群中共轭的算法，并具有更好的理论性质。我们表明，特别是，如果一个超级首脑会议的元素共轭是“刚性”（即，具有一定的特别简单的结构），然后通过共轭的正元素获得这样一个刚性共轭的最佳方式是由迭代循环滑动。
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A New Approach to the Conjugacy Problem in Garside Groups
循环运算赋予Garside群$G$的任意元素$x$的超顶点集$S_x$有向图$Γ_x$的结构。我们建立了由$Γ_x$的电路组成的$S_x$的子集$U_x$可以用来代替$S_x$来决定$G$中的$x$的共轭性，从而更快更实用地解决Garside群的共轭性问题。此外，我们提出了一种概率方法的共轭搜索问题的Garside群。这些结果可能会对最近提出的基于辫群中共轭（搜索）问题的安全性产生影响。
math.GRmath.GT
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On reduction curves and Garside properties of braids
In this paper we study the reduction curves of a braid, and how they can be used to decompose the braid into simpler ones in a precise way, which does not correspond exactly to the decomposition given by Thurston theory. Then we study how a cyclic sliding (which is a particular kind of conjugation) affects the normal form of a braid with respect to the normal forms of its components. Finally, using the above methods, we provide the example of a family of braids whose sets of sliding circuits (hence ultra summit sets) have exponential size with respect to the number of strands and also with res…
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Garside群的共轭I：循环、幂和刚性
Conjugacy in Garside groups I: Cyclings, powers, and rigidity
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Periodic elements in Garside groups
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A Garside-theoretic approach to the reducibility problem in braid groups
令$D_n$表示复平面中的$n$穿孔盘，其中穿孔在实轴上。如果在$D_n$中存在基本曲线系统$\C$，称为$α$的\emph{reduction system}，使得$α*\C=\C$，其中$α*\C$表示编织$α$对曲线系统$\C$的作用，则称$n$ -编织$α$为\emph{reducible}. $D_n$中的曲线系统$\C$称为\emph{standard}，如果它的每个分量都与以实轴为中心的圆同位素。本文研究了将一个曲线系转化为一个标准曲线系的辫子的特征，进而研究了可约辫子的共轭类的特征。对于$D_n$中的本质曲线系统$\C$，定义$\C$的\emph{standardizer}为$\St(\C)=\{P\in B_n^+:P*\C{is standard}\}$，并证明了$\St(\C)$是$B_n^+$的一个子格.特别地，在$\St(\C)$中存在唯一的极小元。利用可约辫的极小元和可约辫的规范归约系统，定义了可约辫的最外分支，并证明了对于最外分支比全辫（包括分裂辫）简单的可约辫，其超顶点集的每个元素都有一个标准归约系统.这意味着，对于这样的辫子，找到一个约简系统就像找到一个超顶点集的单个元素一样容易。
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An element in Artin's braid group $B_n$ is called periodic if it has a power which lies in the center of $B_n$. The conjugacy problem for periodic braids can be reduced to the following: given a divisor $1\le d<n-1$ of $n-1$ and an element $α$ in the super summit set of $ε^d$, find $γ\in B_n$ such that $γ^{-1}αγ=ε^d$, where $ε=(σ_{n-1}\cdotsσ_1)σ_1$. In this article we characterize the elements in the super summit set of $ε^d$ in the dual Garside structure by studying the combinatorics of noncrossing partitions arising from periodic braids. Our characterization directly provides a conjugating …
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A family of pseudo-Anosov braids whose super summit sets grow exponentially
We prove that the size of the super summit set of a braid can grow exponentially with the canonical length of the braid, even for pseudo-Anosov braids.
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