The coarse Novikov conjecture for extensions of coarsely embeddable groups

TL;DR

This paper proves that the coarse Novikov conjecture holds for a sequence of group extensions when both the kernel and quotient groups are coarsely embeddable into Hilbert space, even if the entire group sequence isn't.

Contribution

It establishes the coarse Novikov conjecture for extensions of groups with uniformly bounded geometry, under the condition that the kernel and quotient are coarsely embeddable into Hilbert space.

Findings

The conjecture holds for the group sequence under the given conditions.

The result applies even if the entire group sequence does not embed into Hilbert space.

Provides new insights into the coarse geometric properties of group extensions.

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.