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

Abstract

Let $\left( 1\to N_n\to G_n\to Q_n\to 1 \right)_{n\in \mathbb{N}}$ be a sequence of extensions of finitely generated groups with uniformly finite generating subsets. We show that if the sequence $\left( N_n \right)_{n\in \mathbb{N}}$ with the induced metric from the word metrics of $\left( G_n \right)_{n\in \mathbb{N}}$ has property A, and the sequence $\left( Q_n \right)_{n\in \mathbb{N}}$ with the quotient metrics coarsely embeds into Hilbert space, then the coarse Baum-Connes conjecture holds for the sequence $\left( G_n \right)_{n\in \mathbb{N}}$, 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}