We discuss two results on bi-Lipschitz embeddings of metric spaces. The first is a (negative) answer to a 1997 question of Heinonen and Semmes asking whether spaces that can be “folded” into Euclidean space can be embedded bi-Lipschitzly. The second is a more general theorem showing that spaces with “thick” curve families cannot have bi-Lipschitz embeddings in Euclidean space, unless they admit some “infinitesimal splitting”. This latter result has some applications to the study of conformal dimension. This is joint work with Sylvester Eriksson-Bique (University of Jyvaskyla). 

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