Coxeter群的胖化Davis复形与加权$L^2$ -（上）同调
The fattened Davis complex and weighted $L^2$-(co)homology of Coxeter groups
摘要：
与Coxeter系统$(W,S)$相关联的是一个可收缩的单纯复形$Σ$，称为戴维斯复形，$W$在该复形上适当地起作用并且通过反射而共压缩。给定一个正实数多参数$\mathbf{q}$，可以定义$Σ$的加权$L^2$ -（上）同调群，并将它们与一个称为加权$L^2$ -贝蒂数的非负实数相关联。当$\mathbf{q}$位于某个限制范围之外时，对这些组的行为知之甚少，并且加权$L^2$ -贝蒂数已被证明难以计算。在这篇文章中，我们提出了一个程序来计算加权的$L^2$ -（共）同源性的$Σ$考虑增厚版本的这种复杂的。该程序证明特别成功的加权$L^2$ -（上）同调的某些无限特殊子群的$W$在低维消失。然后，我们使用我们的复杂的Coxeter群的许多例子进行计算，在大多数情况下提供显式公式的加权$L^2$ -贝蒂数。
英文摘要：
Associated to a Coxeter system $(W,S)$ there is a contractible simplicial complex $Σ$ called the Davis complex on which $W$ acts properly and cocompactly by reflections. Given a positive real multiparameter $\mathbf{q}$, one can define the weighted $L^2$-(co)homology groups of $Σ$ and associate to them a nonnegative real number called the weighted $L^2$-Betti number. Not much is known about the behavior of these groups when $\mathbf{q}$ lies outside a certain restricted range, and weighted $L^2$-Betti numbers have proven difficult to compute. In this article we propose a program to compute the weighted $L^2$-(co)homology of $Σ$ by considering a thickened version of this complex. The program proves especially successful provided that the weighted $L^2$-(co)homology of certain infinite special subgroups of $W$ vanishes in low dimensions. We then use our complex to perform computations for many examples of Coxeter groups, in most cases providing explicit formulas for the weighted $L^2$-Betti numbers.
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