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Distortion measures how a mapping changes distances or geometric quantities relative to their original values. For an injective map between metric spaces, one multiplicative measure is the product of its Lipschitz constant and the Lipschitz constant of its inverse on the image. Subgroup distortion instead compares intrinsic and ambient word metrics through a growth function.

For a finitely generated subgroup $H\le G$, subgroup distortion compares the intrinsic word length $|h{|}_H$ with the ambient word length $|h{|}_G$. The subgroup is undistorted exactly when its inclusion is a quasi-isometric embedding.

This is the subgroup word-metric instance of distortion.