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We study metric measure spaces that admit "thick" families of rectifiablecurves or curve fragments, in the form of Alberti representations or curvefamilies of positive modulus. We show that such spaces cannot be bi-Lipschitzembedded into any Euclidean space unless they admit some "infinitesimalsplitting": their tangent spaces are bi-Lipschitz equivalent to product spacesof the form $Z\times \mathbb{R}^k$ for some $k\geq 1$. We also provideapplications to conformal dimension and give new proofs of some previouslyknown non-embedding results.

Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings