Embedding complexity into Banach spaces and the strong Novikov conjecture

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.