论文标题
奇数coxeter组的自动形态
Automorphisms of odd Coxeter groups
论文作者
论文摘要
一个奇数coxeter $ w $是一个承认所有指数$ m_ {ij} $的coxeter系统$（w，s）$的奇数。该论文研究了奇数coxeter组的家族，其关联的标记图$ \ MATHCAL {V} _ {（w，s）} $是树。众所周知，当且仅当他们接受具有相同等级的Coxeter系统和相同的有限指数的Coxeter系统时，该家族中的两个Coxeter组是同构的。特别是，该家族中的每个组都是同构的，该组接纳了一个相关标记的图形的Coxeter系统。我们给出了该组的自动形态组的完整描述，并得出了足够的条件，可以将自动态组分裂为内部和外部自动形态组的半导体产物。作为应用程序，我们证明该家族中的Coxeter组满足$ r_ \ infty $ - property and is（co）-hopfian。我们比较一个特殊的奇数coxeter组的结构属性，自动形态组，$ \ r_ \ infty $ -property和（co） - hopfianity，其唯一有限指数是辫子组和双胞胎组的三个。
An odd Coxeter group $W$ is one which admits a Coxeter system $(W,S)$ for which all the exponents $m_{ij}$ are either odd or infinity. The paper investigates the family of odd Coxeter groups whose associated labeled graphs $\mathcal{V}_{(W,S)}$ are trees. It is known that two Coxeter groups in this family are isomorphic if and only if they admit Coxeter systems having the same rank and the same multiset of finite exponents. In particular, each group in this family is isomorphic to a group that admits a Coxeter system whose associated labeled graph is a star shaped tree. We give the complete description of the automorphism group of this group, and derive a sufficient condition for the splitting of the automorphism group as a semi-direct product of the inner and the outer automorphism groups. As applications, we prove that Coxeter groups in this family satisfy the $R_\infty$-property and are (co)-Hopfian. We compare structural properties, automorphism groups, $\R_\infty$-property and (co)-Hopfianity of a special odd Coxeter group whose only finite exponent is three with the braid group and the twin group.
