There is no equivariant coarse embedding of $L_{p}$ into $\ell_p$

TL;DR

This paper proves that the Banach space $L_{p}$ cannot be coarsely embedded into $ ell_p$ in an equivariant manner, by showing all group representations into isometries are trivial, thus reducing the problem to bi-Lipschitz embeddings.

Contribution

It establishes the non-existence of equivariant coarse embeddings of $L_{p}$ into $ ell_p$, advancing understanding of geometric group theory and Banach space embeddings.

Findings

No proper, affine, isometric action of $L_{p}$ on $ ell_p$ exists.

Representations of $L_{p}$ into isometries of $ ell_p$ are trivial.

The problem reduces to the bi-Lipschitz setting for embeddings.

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.