Coarse selectors of groups

Abstract

For a group \(G\), \(\mathcal{F}_G\) denotes the set of all non-empty finite subsets of \(G\). We extend the finitary coarse structure of \(G\) from \(G\times G\) to \(\mathcal{F}_G\times \mathcal{F}_G\) and say that a macro-uniform mapping \(f\colon \mathcal{F}_G \to \mathcal{F}_G\) (resp. \(f\colon [G]^2 \to G\)) is a finitary selector (resp. 2-selector) of \(G\) if \(f(A)\in A\) for each \(A\in \mathcal{F}_G\) (resp. \( A \in [G]^2 \)). We prove that a group \(G\) admits a finitary selector if and only if \(G\) admits a 2-selector and if and only if \(G\) is a finite extension of an infinite cyclic subgroup or \(G\) is countable and locally finite. We use this result to characterize groups admitting linear orders compatible with finitary coarse structures.