Artin groups and geometric monodromy
Artin groups and geometric monodromy
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Artin群和几何单一性
DOI：
10.1007/s002220050353
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发表时间：
1999
影响因子：
3.1
通讯作者：
B. Wajnryb
中科院分区：
文献类型：
--
作者：
B. Wajnryb
本文的主要结果是：E6，E7，E8型的Artin群（广义辫子群）不能嵌入到曲面的映射类群中，使得Artin群的标准生成元映射到Dehn twist上.文献[8]和[7]中考虑了其他Artin群的类似问题。群Ei是同名孤立奇点的几何单值同态的域，其中目标群是曲面的映射类群。单值同态将Artin群的标准生成元带到Dehn twist上。上述结果否定地回答了Sullivan的一个问题（见[1]）：曲线的孤立奇点的几何单值性是否总是内射的？我们首先回顾一些定义。一个Coxeter群是一个可生成的群G，它允许下面的表示。它由a1，a2，...，ak，其中ni，j是正整数或∞（当aiaj在G中具有无穷阶时），且ni，j= 1当且仅当i= j。已知G中元素aiaj的阶等于ni，j。在存在关系的情况下，
The main result of this paper says that Artin groups (generalized braid groups) of type E6, E7, E8 cannot be embedded into a mapping class group of a surface in such a way that the standard generators of the Artin group are mapped onto Dehn twists. Similar problem for some other Artin groups was considered in papers [8] and [7]. Groups Ei are the domains of geometric monodromy homomorphism of isolated singularities of the same name, where the target group is the mapping class group of a surface. The monodromy homomorphism takes the standard generators of the Artin group onto Dehn twists. The above result gives a negative answer to a question of Sullivan (see [1]): is a geometric monodromy of an isolated singularity of a curve always injective?We recall first some definitions. A Coxeter group is a finitely generated group G which admits the following presentation. It is generated by a1, a2,..., ak with the defining relations (aiaj) ni, j= 1, where ni, j is a positive integer or∞(when aiaj has infinite order in G) and ni, j= 1 if and only if i= j. It is known that the order of the element aiaj in G is equal to ni, j. In the presence of relations