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 $f:{\mathcal{F}}_G\to {\mathcal{F}}_G$ (resp. $f:[G{]}^2\to G$) 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 iff $G$ admits a 2-selector and iff $G$ 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.

finitary coarse structure, Cayley graph, selector