There is no equivariant coarse embedding of $L_p$ into ${\ell }_p$

Abstract

In this paper we prove that $L_p$ does not admit an equivariant coarse embedding into ${\ell }_p$ i.e there is no proper, affine, isometric action of $L_p$, viewed as a group under addition with the standard metric $||.|{|}_p$, on ${\ell }_p$. This is done by showing that representations of $L_p$ into $Isom({\ell }_p)$ has to be trivial, which allows us to reduce the question to bi-Lipschitz setting.

References in corpus (1)

• Euclidean quotients of finite metric spaces