Coarse selectors of groups

TL;DR

This paper extends coarse structures to finite subsets of groups, characterizes groups with selectors, and explores conditions for compatible linear orders, revealing structural group properties.

Contribution

It introduces finitary selectors and characterizes groups that admit such selectors, linking group structure to coarse geometric properties.

Findings

Groups admit finitary selectors iff they admit 2-selectors.

Groups with selectors are finite extensions of infinite cyclic groups or countable locally finite.

Characterizes groups with linear orders compatible with finitary coarse structures.

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 $f: \mathcal{F}_G \rightarrow \mathcal{F}_G$ (resp. $f: [G]^2 \rightarrow G$) 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.