Nous montrons que pour les groupes nilpotents sous-riemanniens, le cône tangent en l’identité n’est pas un invariant quasi-conforme complet. À savoir, nous montrons qu’il existe un groupe de Lie nilpotent muni d’une métrique sous-riemannienne invariante à gauche qui n’est pas localement quasi-conformément équivalent à son cône tangent en l’identité. En particulier, ces espaces ne sont pas localement bi-Lipschitziens. Le résultat repose sur l’étude des groupes de Carnot qui sont rigides dans le sens que leurs seules applications quasi-conformes sont les translations et les dilatations.
We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In particular, such spaces are not locally bi-Lipschitz homeomorphic. The result is based on the study of Carnot groups that are rigid in the sense that their only quasiconformal maps are the translations and the dilations.
@article{AIF_2014__64_6_2265_0,
author = {Le Donne, Enrico and Ottazzi, Alessandro and Warhurst, Ben},
title = {Ultrarigid tangents of sub-Riemannian nilpotent groups},
journal = {Annales de l'Institut Fourier},
volume = {64},
year = {2014},
pages = {2265-2282},
doi = {10.5802/aif.2912},
zbl = {06387339},
mrnumber = {3331166},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2014__64_6_2265_0}
}
Le Donne, Enrico; Ottazzi, Alessandro; Warhurst, Ben. Ultrarigid tangents of sub-Riemannian nilpotent groups. Annales de l'Institut Fourier, Tome 64 (2014) pp. 2265-2282. doi : 10.5802/aif.2912. http://gdmltest.u-ga.fr/item/AIF_2014__64_6_2265_0/
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