On the measure of conformal difference between Euclidean and Lobachevsky spaces

Euclidean space $\mathbb R^n$ and Lobachevsky space $\mathbb H^n$ are known to be not equivalent either conformally or quasiconformally. In this work we give exact asymptotics of the critical order of growth at infinity for the quasiconformality coefficient of a diffeomorphism $f\colon \mathbb R^n\to\mathbb H^n$ for which such a mapping $f$ is possible. We also consider the general case of immersions $f\colon M^n\to N^n$ of conformally parabolic Riemannian manifolds.
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