Lipschitz functions on unions and quotients of metric spaces

Given a finite collection {Xi}i is an element of I of metric spaces, each of which has finite Nagata dimension and Lipschitz free space isomorphic to L1, we prove that their union has Lipschitz free space isomorphic to L1. The short proof we provide is based on the Pelczynski decomposition method. A corollary is a solution to a question of Kaufmann about the union of two planar curves with tangential intersection. A second focus of the paper is on a special case of this result that can be studied using geometric methods. That is, we prove that the Lipschitz free space of a union of finitely many quasiconformal trees is isomorphic to L1. These geometric methods also reveal that any metric quotient of a quasiconformal tree has Lipschitz free space isomorphic to L1. Finally, we analyze Lipschitz light maps on unions and metric quotients of quasiconformal trees in order to prove that the Lipschitz dimension of any such union or quotient is equal to 1.