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 >0 and compact setKG are given, there is a compact setU with 0<|U|< and |x U U|/|U|< for allxK.(A) If >0 and compact setKG, which includes the unit, are given there is a compact setU with 0<|U|< and |K U U|/|U|<.HereA B=(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 compact, there is an indexed family {x J}G such that {Kx } is a covering forG whose covering index at each pointg (the number of J withgKx ) is uniformly bounded throughoutG.