Asymptotic behavior at infinity of the admissible growth of the quasiconformality coefficient and the injectivity of immersions of sub-Riemannian manifolds

We determine the asymptotic behavior of the admissible growth of the quasiconformality coefficient in a general global injectivity theorem for immersions of sub-Riemannian manifolds of conformally parabolic type. In the model case of a contact immersion of the Heisenberg group in itself, the asymptotic behavior of the admissible growth of the quasiconformality coefficient for which the mapping is still globally invertible was found by the author earlier.