Covering properties and Følner conditions for locally compact groups

Summary

LetG be a locally compact group with left Haar measurem _G on the Borel sets IB(G) (generated by open subsets) and write |E|=m _G (E). Consider the following geometric conditions on the groupG.

(FC If ɛ>0 and compact setK⊂G are given, there is a compact setU with 0<|U|<∞ and |x U ΔU|/|U|<ɛ for allxεK.

(A) If ɛ>0 and compact setK⊂G, which includes the unit, are given there is a compact setU with 0<|U|<∞ and |K U ΔU|/|U|<ɛ.

HereA ΔB=(A/B)⌣(B/A) is the symmetric difference set; by regularity ofm _G it makes no difference if we allowU to be a Borel set. It is well known that (A)⇒(FC) and it is known that validity of these conditions is intimately connected with “amenability” ofG: the existence of a left invariant mean on the spaceCB(G) of all continuous bounded functions. We show, for arbitrary locally compact groupsG, that (amenable)⇔(FC)⇔(A). The proof uses a covering property which may be of interest by itself: we show that every locally compact groupG satisfies.

(C) For at least one setK, with int(K)≠Ø and$\bar K$ compact, there is an indexed family {x _α∶αεJ}⊂G such that {Kx _α} is a covering forG whose covering index at each pointg (the number of αεJ withgεKx _α) is uniformly bounded throughoutG.