Approximation properties in Lipschitz-free spaces over groups

TL;DR

This paper investigates the approximation properties of Lipschitz-free spaces over groups with certain regularity, showing they have the metric approximation property and Schauder bases in various cases, including hyperbolic and Artin groups.

Contribution

It establishes the metric approximation property and existence of Schauder bases for Lipschitz-free spaces over groups with invariant metrics, extending previous results to new classes of groups.

Findings

Lipschitz-free spaces over compact metrizable groups have the metric approximation property.

Certain groups admit Lipschitz-free spaces with Schauder bases, including hyperbolic and Artin groups.

Examples include nets in hyperbolic spaces, which also have Schauder bases.

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 $G$ equipped with an arbitrary compatible left-invariant metric $d$, the Lipschitz-free space over $G$, $\mathcal{F}(G,d)$, satisfies the metric approximation property. We show also that, given a finitely generated group $G$, with its word metric $d$, from a class of groups admitting a certain special type of combing, which includes all hyperbolic groups and Artin groups of large type, $\mathcal{F}(G,d)$ has a Schauder basis. Examples and applications are discussed. In particular, for any net $N$ in a real hyperbolic $n$-space $\mathbb{H}^n$, $\mathcal{F}(N)$ has a Schauder basis.