On the weak-fragmentability index of some Lipschitz-free spaces

Abstract: We show the existence of Lipschitz-free spaces verifying the Point of Continuity Property with arbitrarily high weak-fragmentability index. For this purpose, we use a generalized construction of the countably branching diamond graphs. As a consequence, we deduce that to be Lipschitz-universal for countable complete metric spaces, a separable complete metric space cannot be purely 1-unrectifiable. Another corollary is the existence of an uncountable family of pairwise non-isomorphic Lipschitz-free spaces over purely 1-unrectifiable metric spaces. Some results on compact reduction are also obtained.