有限群的合成建筑
Synthetic Buildings for Finite Groups
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中文总结 AI 辅助
本文为有限群$G$和素数$p$引入合成建筑，研究其类型、极小性、维数及同伦型等性质，提出相关猜想。
中文摘要 AI 辅助
我们为每个有限群$G$和素数$p$引入合成建筑。这些是$G$-单纯复形，其中包括K.S. Brown的$p$-子群复形，以及特征$p$下李型有限群的建筑。合成建筑因其特殊性质而有用，包括提供$G$的上同调公式。我们的目标是描述可能出现的合成建筑类型，并理想地对其进行分类。极小合成建筑发挥特殊作用，对于大多数群而言，这类建筑数量不多。不过，我们证明可以找到有限群序列，随着序列推进，其极小合成建筑数量不断增加，还能找到具有不同维数的极小合成建筑的群。也能找到在给定维数下具有无限多个非极小合成建筑的群。在其他情形中，我们证明不存在等变同伦型不同于Brown复形和单点复形的极小合成建筑。我们提出了若干猜想。
英文摘要
We introduce synthetic buildings for each finite group $G$ and prime $p$. These are $G$-simplicial complexes, among which are the $p$-subgroups complex of K.S. Brown, and also the buildings of finite groups of Lie type in characteristic $p$. Synthetic buildings are useful because of special properties, including providing a formula for the cohomology of $G$. Our goal is to describe the kinds of synthetic buildings that can arise, and ideally to classify them. The \textit{minimal} synthetic buildings play a special role and for most groups there are not very many of these. However, we show that it is possible to find sequences of finite groups with increasingly many minimal synthetic buildings as we move along the sequence, and also groups with minimal synthetic buildings of different dimensions. It is also possible to find groups with infinitely many non-minimal synthetic buildings of a given dimension. In other situations, we show that there are no minimal synthetic buildings of equivariant homotopy type distinct from those of Brown's complex and the complex consisting of a single point. We make a number of conjectures.
