特征、$L^2$ -贝蒂数和一个等变逼近定理
Characters, $L^2$-Betti numbers and an equivariant approximation theorem
摘要：
设$G$是具有有限子群$H$的群。定义了一个真$G$ -CW-复形的$L^2$ -同调中$H$的不可约表示的$L^2$ -重数.这些不变量推广了$L^2$ -贝蒂数。我们的主要结果是$L^2$ -重数的逼近定理，它们分别推广了Lück，Farber和Elek-Szabó关于$L^2$ -贝蒂数的逼近定理.主要内容是无限群的特征标理论和由有限子群导出特征标的方法。我们讨论应用（算术）群的上同调。
英文摘要：
Let $G$ be a group with a finite subgroup $H$. We define the $L^2$-multiplicity of an irreducible representation of $H$ in the $L^2$-homology of a proper $G$-CW-complex. These invariants generalize the $L^2$-Betti numbers. Our main results are approximation theorems for $L^2$-multiplicities which extend the approximation theorems for $L^2$-Betti numbers of Lück, Farber and Elek-Szabó respectively. The main ingredient is the theory of characters of infinite groups and a method to induce characters from finite subgroups. We discuss applications to the cohomology of (arithmetic) groups.
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