Coarse embeddings at infinity and generalized expanders at infinity

TL;DR

This paper introduces a new concept of coarse embedding at infinity for metric spaces, linking group properties to embeddings of associated box spaces, and identifies generalized expanders at infinity as obstructions.

Contribution

It defines coarse embedding at infinity, relates it to group box spaces, and introduces generalized expanders at infinity as obstructions to such embeddings.

Findings

A residually finite group admits a coarse embedding into Hilbert space iff its box space does.

Generalized expanders at infinity obstruct coarse embeddability.

The notion generalizes Gromov's coarse embedding concept.

Abstract

We introduce a notion of coarse embedding at infinity into Hilbert space for metric spaces, which is a weakening of the notion of fibred coarse embedding and a far generalization of Gromov's concept of coarse embedding. It turns out that a residually finite group admits a coarse embedding into Hilbert space if and only if one (or equivalently, every) box space of the group admits a coarse embedding at infinity into Hilbert space. Moreover, we introduce a concept of generalized expander at infinity and show that it is an obstruction to coarse embeddability at infinity.