A finitely-generated amenable group with very poor compression into Lebesgue spaces

TL;DR

This paper constructs a finitely-generated amenable group that cannot be coarsely embedded into Lebesgue spaces with positive compression, answering a significant open question in geometric group theory.

Contribution

It provides the first example of such a group, demonstrating limitations of coarse embeddings into L_p spaces for amenable groups.

Findings

No coarse 1-Lipschitz embedding with positive compression into L_p exists for the constructed group.

Answers an open question by Arzhantseva, Guba, and Sapir.

Highlights the complexity of embedding properties of amenable groups.

Abstract

We construct an example of a finitely-generated amenable group that does not admit any coarse 1-Lipschitz embedding with positive compression exponent into L_p for any 1 \leq p < \infty, answering positively a question of Arzhantseva, Guba and Sapir.