The coarse Baum-Connes conjecture for certain extensions and relative expanders

Let $left( 1to N_nto G_nto Q_nto 1 right)_{nin mathbb{N}}$ be a sequence of extensions of finitely generated groups with uniformly finite generating subsets. We show that if the sequence $left( N_n right)_{nin mathbb{N}} $ with the induced metric from the word metrics of $left( G_n right)_{nin mathbb{N}} $ has property A, and the sequence $left( Q_n right)_{nin mathbb{N}} $ with the quotient metrics coarsely embeds into Hilbert space, then the coarse Baum-Connes conjecture holds for the sequence $left( G_n right)_{nin mathbb{N}}$, which may not admit a coarse embedding into Hilbert space. It follows that the coarse Baum-Connes conjecture holds for the relative expanders and group extensions exhibited by G. Arzhantseva and R. Tessera, and special box spaces of free groups discovered by T. Delabie and A. Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embedded expander. This in particular solves an open problem raised by G. Arzhantseva and R. Tessera cite{Arzhantseva-Tessera 2015}.