Abstract
We compare the Grushin geometry to Euclidean geometry, through quasisymmetric parametrization, bilipschitz parametrization and bilipschitz embedding, highlighting the role of the exponents and the fractal nature of the singular hyperplanes in Grushin geometry. Consider in Rn a system of diagonal vector fields Xj = λj(x) ∂ ∂xj , j = 1, . . . , n,
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