论文标题
ARTIN MONOIDS的Deligne综合体
A Deligne complex for Artin Monoids
论文作者
论文摘要
在本文中，我们介绍并研究了一些与Artin Monoid相关的几何对象。 Artin组的Deligne复合物是第二作者和戴维斯（Davis，1995）引入的立方体复合体，以研究这些组的K（π，1）猜想。使用Artin Monoid Coset的概念，我们为Artin Monoids构建了Deligne Complex的版本。我们表明，对于任何Artin Monoid，该立方体综合体都是可缩度的。此外，我们研究了相应的ARTIN组中Monoid Deligne复合物中的嵌入到Deligne复合物中。我们表明，对于任何Artin组，这都是当地等距嵌入。在FC型Artin组的情况下，可以将此结果加强到全球等距嵌入，因此，Monoid Deligne复合物是CAT（0），其在Deligne复合物中的图像是凸的。我们还考虑了Artin组的Cayley图，并研究了Artin Monoid元素跨越子图的性质。我们的最终结果表明，对于有限型Artin组，Monoid Cayley图将异图像嵌入（而不是准串联）嵌入到Cayley图组中。
In this paper we introduce and study some geometric objects associated to Artin monoids. The Deligne complex for an Artin group is a cube complex that was introduced by the second author and Davis (1995) to study the K(π,1) conjecture for these groups. Using a notion of Artin monoid cosets, we construct a version of the Deligne complex for Artin monoids. We show that for any Artin monoid this cube complex is contractible. Furthermore, we study the embedding of the monoid Deligne complex into the Deligne complex for the corresponding Artin group. We show that for any Artin group this is a locally isometric embedding. In the case of FC-type Artin groups this result can be strengthened to a globally isometric embedding, and it follows that the monoid Deligne complex is CAT(0) and its image in the Deligne complex is convex. We also consider the Cayley graph of an Artin group, and investigate properties of the subgraph spanned by elements of the Artin monoid. Our final results show that for a finite type Artin group, the monoid Cayley graph embeds isometrically, but not quasi-convexly, into the group Cayley graph.
