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

TL;DR

This work develops a deep link between Cohen--Lyndon quotients and equivariant homology to address the Baum–Connes conjecture in new settings. It constructs Dehn filling spaces and proper-action Dehn filling complexes that support excision in equivariant homology and analytic K-homology, enabling explicit diagrams and K-theory calculations for Cohen--Lyndon aspherical groups. The authors prove the Baum–Connes conjecture with coefficients with finite wreath products for broad classes, including all hyperbolic groups and, via relative hyperbolicity, many relatively hyperbolic groups, with permanence under extensions. A key technical contribution is a gamma-element based passage to wreath products, yielding BCC_wr for wreath products of hyperbolic groups and preservation under finite extensions. The results have concrete applications to complete finite-volume negatively curved manifolds, lattices in rank-one Lie groups, Einstein metrics on Dehn-filled manifolds, and quotients of mapping class groups, significantly widening the scope of BC-type results in geometric group theory and noncommutative geometry.

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.

Figures (1)

• Figure 1: Dehn filling complex for proper actions associated to a Cohen--Lyndon triple $(G,\mathcal{H},\{N_{\lambda}\}_{\lambda\in\Lambda})$. The left and middle circles represent the $\overline{G}$-translates of spaces corresponding to a single subgroup $H_{\lambda} \in \mathcal{H}$, while the dots indicate the $\overline{G}$-translates of spaces corresponding to all other subgroups in $\mathcal{H}$.

Theorems & Definitions (113)

• Theorem 1.1: Thurston thurston1983three
• Theorem 1: Theorem \ref{['thm_CLP_diagram']}
• Definition 1.2: The Baum–Connes Conjecture with Coefficients with finite wreath products
• Theorem 2: Theorem \ref{['thm_BCC_rel']}
• Theorem 3: Theorem \ref{['thm_hyp_BCC_wr']}
• Theorem 4: Theorem \ref{['thm_gamma_wr']}
• Corollary 5: Corollary \ref{['cor_gamma_to_wreath']}
• Corollary 6: Corollary \ref{['cor_negative_curvature']}
• Corollary 7: Corollary \ref{['cor_lattice_BCC']}
• Corollary 8: Corollary \ref{['cor_Dehn_filling_mfld']}