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

Abstract

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.