AI 中文总结
研究虚拟阿廷群，通过构建德利涅复形类似物对其展开几何研究，证明对局部可约定义图是CAT(0)，并应用于有限子群分类及证明$K(\pi, 1)$猜想类似物，得出群几乎无挠且有最小维度余紧分类空间模型等结论。
AI 中文摘要
虚拟阿廷群由贝林热里 - 巴黎 - 蒂尔最近作为虚拟辫群的推广引入。本文通过构建虚拟阿廷群的德利涅复形类似物对这些群展开几何研究，证明对于所有局部可约定义图（特别包含二维图和无任何标签3的图，且在戈德堡 - 瓦斯科意义下是一般的）它是CAT(0)。作为应用，对局部可约虚拟阿廷群的有限子群进行分类，表明此类群共轭到相应考克斯特子群的同构副本中。还证明了局部可约虚拟阿廷群的$K(\pi, 1)$猜想类似物：这些群几乎无挠且允许具有最小维度的恰当作用分类空间的余紧模型，该维度等于群的虚拟上同调维数。
英文摘要
Virtual Artin groups were recently introduced by Bellingeri-Paris-Thiel as a generalisation of virtual braid groups. In this article, we initiate a geometric study of these groups by constructing an analogue of the Deligne complex for virtual Artin groups, and we prove that it is CAT(0) for all locally reducible defining graphs (a class that contains in particular two-dimensional graphs and graphs without any label $3$, and which is generic in the sense of Goldsborough-Vaskou). As applications, we classify finite subgroups of locally reducible virtual Artin groups, showing that such groups are conjugated into an isomorphic copy of the corresponding Coxeter subgroup. We also prove an analogue of the $K(π, 1)$-conjecture for locally reducible virtual Artin groups: we show that these groups are virtually torsion-free and admit a cocompact model of classifying space for proper actions of minimal dimension, equal to the virtual cohomological dimension of the group.
Comments41 pages, 18 figures
