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. We determine the structure of the commutator subgroups of groups in this family. 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. As a consequence, it follows that the Coxeter groups in this family satisfy the $R_\infty$-property and are (co)-Hopfian. We conclude with a comparison of a special odd Coxeter group whose only finite exponent is three with the braid group and the twin group.

中文翻译：

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奇 Coxeter 群的自同构

奇数 Coxeter 群 $W$ 允许 Coxeter 系统 $(W,S)$ 的所有指数 $m_{ij}$ 要么是奇数要么是无穷大。该论文研究了奇数 Coxeter 群家族，其关联的标记图 $\mathcal{V}_{(W,S)}$ 是树。我们确定这个族中群的交换子子群的结构。已知该族中的两个 Coxeter 群是同构的，当且仅当它们承认 Coxeter 系统具有相同的秩和相同的有限指数多重集。特别地，这个族中的每个群与一个接纳 Coxeter 系统的群同构，该系统的相关标记图是一棵星形树。我们给出了这个群的自同构群的完整描述。因此，该族中的 Coxeter 群满足 $R_\infty$-性质并且是 (co)-Hopfian。