Sufficiency of previously known permanence for BCC on products of hyperbolic groups

Ascertain whether the Baum–Connes conjecture with coefficients for all groups that are commensurable to a finite product of hyperbolic groups follows solely from previously known permanence properties together with Lafforgue’s proof of the Baum–Connes conjecture for hyperbolic groups, without using the new wreath-product results developed here.

Background

The authors prove that hyperbolic groups satisfy the Baum–Connes conjecture with coefficients with finite wreath products and derive broad permanence results, implying the conjecture for groups commensurable to products of hyperbolic groups.

They remark that, prior to their work, it was not clear whether this conclusion already followed from previously known permanence theorems combined with Lafforgue’s theorem for hyperbolic groups, highlighting a gap in the literature regarding the sufficiency of earlier methods.

References

To the best of our knowledge, the BCC for all groups that are commensurable to a product of hyperbolic groups, which is implied by Theorem \ref{intro_thm_hyp_BCC_wr} and the permanence properties (see Section \ref{sec: permanence}), had remained open in general, and it is not clear if it is implied by the previously-known permanence theorems and the BCC for hyperbolic groups (Lafforgue, ).

— Dehn fillings, equivariant homology, and the Baum-Connes conjecture  (2509.15070 - Nishikawa et al., 18 Sep 2025) in Introduction, paragraph following Intro Theorem \ref{intro_thm_hyp_BCC_wr}