In this paper, we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum–Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the $K$-theory of the reduced $C^{\ast }$-algebra of the group. We show the power of this method giving explicit computations for the group ${\mathrm{\text{SL}}}_2(\mathbb{Z}[i])$. In order to carry out these computations we use an Atiyah–Segal type spectral sequence together with the Bredon homology of the classifying space for proper actions.

在本文中，我们提供了一个研究作用于算术离散群的 Bredon（上）同调的 Hecke 算子的框架。我们的主要兴趣在于研究 Bianchi 群的 Hecke 算子。使用 Baum-Connes 猜想，我们可以将计算转移到 Bredon 同调中，以获得对$K$- 减少理论$C^{*}$-群的代数。我们展示了该方法为组提供显式计算的强大功能${\mathrm{\text{SL}}}_2（\mathbb{Z}[我]）$。为了进行这些计算，我们使用 Atiyah-Segal 型谱序列以及分类空间的 Bredon 同源性来进行适当的操作。