We consider a strictly developable simple complex of finite groups $G(\mathcal Q)$. We show that Bestvina's construction for Coxeter groups applies in this more general setting to produce a complex that is equivariantly homotopy equivalent to the standard development. When $G(\mathcal Q)$ is non-positively curved, this implies that the Bestvina complex is a cocompact classifying space for proper actions of $G$ of minimal dimension. As an application, we show that for groups that act properly and chamber transitively on a building of type $(W, S)$, the dimension of the associated Bestvina complex is the virtual cohomological dimension of $W$. We give further examples and applications in the context of Coxeter groups, graph products of finite groups, locally 6-large complexes of groups and groups of rational cohomological dimension at most one. Our calculations indicate that, because of its minimal cell structure, the Bestvina complex is well-suited for cohomological computations.

中文翻译：

---

Bestvina复合体，具有严格的基本领域的团体行动

我们考虑一个有限群$ G（\ mathcal Q）$的可严格展开的简单复数。我们证明Bestvina针对Coxeter小组的构建在这种更一般的情况下适用，从而产生了一个同等的同构同等体，相当于标准开发的复合体。当$ G（\ mathcal Q）$非正弯曲时，这意味着Bestvina复合物是一个紧凑的分类空间，用于最小尺寸的$ G $的适当动作。作为一个应用程序，我们显示出对于在$（W，S）$类型建筑物上正常运行并传递房间的组，关联的Bestvina复合体的维数是$ W $的虚拟同调维数。我们在Coxeter群，有限群的图形乘积，局部6大群的复合体和有理同调维度的群中给出更多示例和应用。