# Asymptotic coarse Lipschitz equivalence

## TL;DR

This paper introduces asymptotic coarse Lipschitz equivalence, a weaker relation than coarse Lipschitz equivalence, and explores its effects on the asymptotic dimension and properties of Banach spaces, including stability results.

## Contribution

It defines asymptotic coarse Lipschitz equivalence, studies its implications on asymptotic dimension, and proves stability of Banach space properties under this relation.

## Key findings

- Asymptotic coarse Lipschitz equivalence is strictly weaker than coarse Lipschitz equivalence.
- Linearly isomorphic to ll_p for 2  p <   p spaces are stable under this equivalence.
- Stability of asymptotic uniform smoothness properties of Banach spaces is established.

## Abstract

We introduce the notion of asymptotic coarse Lipschitz equivalence of metric spaces. We show that it is strictly weaker than coarse Lipschitz equivalence. We study its impact on the asymptotic dimension of metric spaces. Then we focus on Banach spaces. We prove that, for $2\leq p<\infty$, being linearly isomorphic to $\ell_p$ is stable under asymptotic coarse Lipschitz equivalences. Finally, we establish a version of the Gorelik principle in this setting and apply it to prove the stability of various properties of asymptotic uniform smoothness of Banach spaces under asymptotic coarse Lipschitz equivalences.