Coarse selectors of groups

Abstract

For a group G, FG denotes the set of all non-empty finite subsets of G. We extend the finitary coarse structure of G from G×G to FG×FG and say that a macro-uniform mapping f:FG→FG (resp. f:[G]2→G) is a finitary selector (resp. 2-selector) of G if f(A)∈A for each A ∈ FG (resp. A∈[G]2). Weprove 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.