Embedding complexity into Banach spaces and the strong Novikov conjecture

TL;DR

This paper proves that the strong Novikov conjecture holds for discrete groups that can be coarsely embedded with finite complexity into a specific universal Banach space, linking embedding properties to topological conjectures.

Contribution

It establishes a connection between coarse embeddability with finite complexity into a universal Banach space and the validity of the strong Novikov conjecture for discrete groups.

Findings

Strong Novikov conjecture holds for groups with finite complexity coarse embeddings.

Universal Banach space used for embedding is the $igoplus_{p=1}^{ olinebreak} olinebreak ext{ell}^{2p}( olinebreak ext{N})$ space.

Addresses a question posed by Brown-Guentner and Haagerup-Przybyszewska.

Abstract

Brown-Guentner and Haagerup-Przybyszewska showed that every discrete group admits a proper affine isometric action on the universal Banach space $\bigoplus_{p=1}^{\infty} \ell^{2p}(\mathbb{N}),$ taken as the $\ell^{2}$-direct sum, and hence admits a coarse embedding into this space [7, 28]. They further asked whether such embeddings could be used to study the Novikov conjecture. In this paper, we address this question by proving that the strong Novikov conjecture holds for any discrete group that admits a coarse embedding with finite complexity into this universal Banach space.