The coarse Baum–Connes conjecture for certain relative expanders

Let (1→Nm→Gm→Qm→1)m∈N be a sequence of extensions finite groups such that their coarse disjoint unions have bounded geometry. In this paper, we show if the (Nm)m∈N and (Qm)m∈N are coarsely embeddable into Hilbert space, then Baum–Connes conjecture holds for union (Gm)m∈N. As an application, relative expanders constructed by G. Arzhantseva R. Tessera, special box spaces free discovered T. Delabie A. Khukhro, which do not embed yet contain weakly embedded expander. This enlarges class metric known to satisfy conjecture. particular, it solves open problem raised Tessera on expanders.