Building weight-free F{\o}lner sets for Yu's Property A in coarse geometry

TL;DR

This paper investigates conditions under which F{ }lner sets demonstrating Yu's Property A can be chosen within the space itself, providing new insights for various classes of metric spaces and groups.

Contribution

It establishes that for many spaces with Property A, F{ }lner sets can be selected as subsets of the space, including certain groups and box spaces.

Findings

F{ }lner sets can be chosen within the space for spaces with unbounded components

Applicable to all countable discrete groups with Property A

Works for all box spaces with Property A

Abstract

In this note we study the natural question of when the generalised F{\o}lner sets exhibiting property A can be chosen to be subsets of the space itself. We show that for many property A spaces $X$, this is indeed possible. Specifically this holds: for all discrete bounded geometry metric spaces which coarsely have all components unbounded; for all countable discrete groups; and for all box spaces.