论文标题
组的简单复合物的同时和几何不变性
Cohomological and geometric invariants of simple complexes of groups
论文作者
论文摘要
我们调查了具有任意地方群体或等效的群体行动的严格开发的简单复合物，承认严格的基本领域。我们介绍了一种计算此类群体的共同体学的新方法。我们还概括了BestVina的结构，以获得与最低尺寸的标准发展相同的多面体复合物。 作为应用程序，对于在$（W，s）$的建筑物建筑物上进行的集体行动，我们表明其Bredon的共同体学维度等于$ W $的虚拟共同体学维度，并实现了建筑物最低维度的建筑。 我们介绍了类似反射的动作的概念，并使用它为布朗在虚拟共同体学维度的平等和布雷登共同体学方面的相等性的强烈猜想中提供了新的反示例家族。 我们表明，简单组的基本组$ g $在树上起作用，稳定器与稳定器一起产生一个子群体$ \ Mathcal {f} $，并且仅当其Bredon共同体学维度相对于$ \ Mathcal {f} $。这证实了一个民间传说的猜想，假设是一个分类空间的模型$ e _ {\ Mathcal {f}} g $ $ g $的$ g $的$ \ nathcal {f} $具有严格的基本域。 为了处理由任意群体行为引起的组的复合物，我们定义了许多组合不变的诸如块poset等组合，这可能具有独立的兴趣。我们还得出了Bredon共同体学维度的一般公式，用于$ e _ {\ Mathcal {f}} g $的组合模型。两者的结果是，我们获得了一个简单的公式，用于$ \ mathrm {cat}（0）$组的适当共同体学维度，其行动承认严格的基本领域。
We investigate strictly developable simple complexes of groups with arbitrary local groups, or equivalently, group actions admitting a strict fundamental domain. We introduce a new method for computing the cohomology of such groups. We also generalise Bestvina's construction to obtain a polyhedral complex equivariantly homotopy equivalent to the standard development of the lowest possible dimension. As applications, for a group acting chamber transitively on a building of type $(W,S)$, we show that its Bredon cohomological dimension is equal to the virtual cohomological dimension of $W$ and give a realisation of the building of the lowest possible dimension. We introduce the notion of a reflection-like action, and use it to give a new family of counterexamples to the strong form of Brown's conjecture on the equality of virtual cohomological dimension and Bredon cohomological dimension for proper actions. We show that the fundamental group $G$ of a simple complex of groups acts on a tree with stabilisers generating a family of subgroups $\mathcal{F}$ if and only if its Bredon cohomological dimension with respect to $\mathcal{F}$ is at most one. This confirms a folklore conjecture under the assumption that a model for the classifying space $E_{\mathcal{F}}G$ of $G$ for the family $\mathcal{F}$ has a strict fundamental domain. In order to handle complexes of groups arising from arbitrary group actions, we define a number of combinatorial invariants such as the block poset, which may be of independent interest. We also derive a general formula for Bredon cohomological dimension for a group $G$ admitting a cocompact model for $E_{\mathcal{F}}G$. As a consequence of both, we obtain a simple formula for proper cohomological dimension of $\mathrm{CAT}(0)$ groups whose actions admit a strict fundamental domain.
