Lipschitz quotients are a nonlinear generalization of linear quotients. A result by Johnson, Lindenstrauss, Preiss and Schechtman states that all certain tree spaces called $\ell_1$-trees are universal 1-Lipschitz quotients for all complete separable geodesic metric spaces. We extend their result by constructing a separable metric space that is a universal 1-Lipschitz quotient for all complete separable metric spaces. We then turn our attention to the compact case and show that there is no compact universal Lipschitz quotient space for all compact metric spaces. To achieve these results, we study curve-flat functions, their connections to Lipschitz quotients on compact spaces, and construct compact metric spaces with high order curve-flat indexes.

All presented results are a joint work with Andrés Quilis and are a combination of new unpublished results and new results obtained in the recent preprint https://arxiv.org/abs/2603.20177.