Characterization of the Haagerup property for residually amenable groups

TL;DR

This paper characterizes residually amenable groups with the Haagerup property through the existence of a fibred cofinitely-coarse embedding of their box families into Hilbert spaces, generalizing previous results for residually finite groups.

Contribution

It introduces the concepts of box family and fibred cofinitely-coarse embedding and extends the characterization of the Haagerup property to residually amenable groups.

Findings

Residually amenable groups with the Haagerup property have a box family admitting a fibred cofinitely-coarse embedding into a Hilbert space.

Generalizes previous results from residually finite groups to residually amenable groups.

Provides a new geometric criterion for the Haagerup property in this broader class of groups.

Abstract

The notions of a box family and fibred cofinitely-coarse embedding are introduced. The countable, residually amenable groups satisfying the Haagerup property are then characterized as those possessing a box family that admits a fibred cofinitely-coarse embedding into a Hilbert space. This is a generalization of a result of X. Chen, Q. Wang and X. Wang on residually finite groups.