论文标题
晶格，Garside结构和弱模块化图
Lattices, Garside structures and weakly modular graphs
论文作者
论文摘要
在本文中，我们研究了各种简单络合物的组合非阳性曲率方面，以及天然的$ \ widetilde a_n $形状的简单，包括$ \ widetilde a_n $的欧几里得建筑物和cayley garside群体的欧几里得建筑物以及garside elements的garside群体。所有这些示例都符合订单增加的$ \ Mathbb z $ - actions和第一名命名作者在先前工作中提出的相关晶格商的更通用环境。我们表明，晶格的标和晶格本身都会产生弱模块化图，这是一种组合非阳性曲率的一种形式。我们还表明，其他几个复合物适合这种晶格/晶格标，因此我们的结果适用，包括$ \ \\ widetilde a_n $的Artin-tits组的Artin Complexses，一类ARC络合物和弱的Garside组，由Bessis的含义是由类似的Garside结构引起的。在此过程中，我们还阐明了分类Garside结构，具有$ \ Mathbb Z $ Action的晶格与研究了本文的不同类别的综合体之间的关系。我们使用此观点来描述具有外来特性的Garside组的第一个例子，例如非线性或刚性结果。
In this article we study combinatorial non-positive curvature aspects of various simplicial complexes with natural $\widetilde A_n$ shaped simplicies, including Euclidean buildings of type $\widetilde A_n$ and Cayley graphs of Garside groups and their quotients by the Garside elements. All these examples fit into the more general setting of lattices with order-increasing $\mathbb Z$-actions and the associated lattice quotients proposed in a previous work by the first named author. We show that both the lattice quotients and the lattices themselves give rise to weakly modular graphs, which is a form of combinatorial non-positive curvature. We also show that several other complexes fit into this setting of lattices/lattice quotients, hence our result applies, including Artin complexes of Artin-Tits groups of type $\widetilde A_n$, a class of arc complexes and weak Garside groups arising from a categorical Garside structure in the sense of Bessis. Along the way, we also clarify the relationship between categorical Garside structure, lattices with $\mathbb Z$ action and different classes of complexes studied this article. We use this point of view to describe the first examples of Garside groups with exotic properties, like non-linearity or rigidity results.
