Approximation properties in Lipschitz-free spaces over groups

Abstract

We study Lipschitz-free spaces over compact and uniformly discrete metric spaces enjoying certain high regularity properties — having group structure with left-invariant metric. Using methods of harmonic analysis we show that, given a compact metrizable group (Formula presented.) equipped with an arbitrary compatible left-invariant metric (Formula presented.), the Lipschitz-free space over (Formula presented.), (Formula presented.), satisfies the metric approximation property. We show also that, given a finitely generated group (Formula presented.), with its word metric (Formula presented.), from a class of groups admitting a certain special type of combing, which includes all hyperbolic groups and Artin groups of large type, (Formula presented.) has a Schauder basis. Examples and applications are discussed. In particular, for any net (Formula presented.) in a real hyperbolic (Formula presented.) -space (Formula presented.), (Formula presented.) has a Schauder basis.