The coarse Novikov conjecture for extensions of coarsely embeddable groups

Abstract

Let $(1\to N_n\to G_n\to Q_n \to 1)_{n\in \mathbb{N}}$ be a sequence of extensions of countable discrete groups. Endow $(G_n)_{n\in \mathbb{N}}$ with metrics associated to proper length functions on $(G_n)_{n\in \mathbb{N}}$ respectively such that the sequence of metric spaces $(G_n)_{n\in \mathbb{N}}$ have uniform bounded geometry. We show that if $(N_n)_{n\in \mathbb{N}}$ and $(Q_n)_{n\in \mathbb{N}}$ are coarsely embeddable into Hilbert space, then the coarse Novikov conjecture holds for the sequence $(G_n)_{n\in \mathbb{N}}$, which may not admit a coarse embedding into Hilbert space.