Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings

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