, X n ) is g-Følner if and only if µ (Xn) = µ (g ?1 Xn) if and only if the shift by g is an isometry

, X n ) is Følner if and only if every shift is an isometry

, If G is finitely generated, then G is amenable if and only if there exists a nondecreasing sequence (X n ) of finite subsets of G such that every shift is an isometry

, Note that one implication of Point 3 was already stated in [Cap09, Theorem 3.5], but the proof contains a confusion between left and right Følner. The full equivalence generalizes

, If (X n ) is the sequence of balls with respect to some generating set of cardinality ?, then every shift is ?-Lipschitz

, G, a nondecreasing exhaustive sequence is g-Følner if and only if all of its subsequences yield a Besicovitch pseudodistance for which the shift by g is continuous

, G is amenable if and only if it admits a nondecreasing exhaustive sequence of finite subsets of which all subsequences yield a Besicovitch distance for which every shift is continuous

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