标题： 德恩填充、等变同调和巴尔-康尼斯猜想 显示英文标题

标题： Dehn fillings, equivariant homology, and the Baum-Connes conjecture

摘要： 我们将Cohen-Lyndon三元组与等变同调理论建立了联系，重点在于Baum-Connes猜想。 在本研究的第一部分，我们建立了用于适当作用的分类空间在等变同调理论中的切除序列。 这为Cohen-Lyndon三元组与Baum-Connes猜想的左侧提供了直接的联系。 独立于这些工作，我们证明了带有系数的Baum-Connes猜想（BCC）与有限幂积对于所有离散双曲群成立，这是基于Lafforgue的开创性工作。 结合这一结果与保持性质以及Dahmani-Guirardel-Osin关于相对双曲群的工作，我们确定了一个广泛的群类，包括所有满足带有有限幂积的BCC的实秩为一的简单李群中的格子。 这显著扩展了我们第一个结果的适用范围，因为 Cohen-Lyndon三元组自然出现在相对双曲群的背景下，从而连接了Baum-Connes猜想的两侧。 显示英文摘要

摘要： We establish a connection between Cohen-Lyndon triples and equivariant homology theory, with a focus on the Baum-Connes conjecture. In the first part of this work, we establish an excision sequence for the classifying spaces for proper actions in equivariant homology theories. This provides a direct link between Cohen-Lyndon triples and the left-hand side of the Baum-Connes conjecture. Independently of these, we prove that the Baum-Connes conjecture with coefficients (BCC) with finite wreath products holds for all discrete hyperbolic groups, building on the monumental work of Lafforgue. Combining this with permanence properties and the work of Dahmani-Guirardel-Osin on relatively hyperbolic groups, we identify a broad class of groups, including all lattices in simple Lie groups of real rank one that satisfy the BCC with finite wreath products. This significantly broadens the scope of our first result, as Cohen-Lyndon triples arise naturally in the context of relatively hyperbolic groups, thereby connecting both sides of the Baum-Connes conjecture.