[Tangents ultra-rigides des groupes nilpotents sous-riemanniens]
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.
Keywords: Sub-Riemannian geometry, metric tangents, Gromov-Hausdorff convergence, nilpotent groups, Carnot groups, quasiconformal maps
Mot clés : Géométrie sous-riemannienne, tangentes métriques, convergence de Gromov-Hausdorff, groupes nilpotents, groupes de Carnot, applications quasi-conforme
@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},
pages = {2265--2282},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {64},
number = {6},
year = {2014},
doi = {10.5802/aif.2912},
zbl = {06387339},
mrnumber = {3331166},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2912/}
}
TY - JOUR AU - Le Donne, Enrico AU - Ottazzi, Alessandro AU - Warhurst, Ben TI - Ultrarigid tangents of sub-Riemannian nilpotent groups JO - Annales de l'Institut Fourier PY - 2014 SP - 2265 EP - 2282 VL - 64 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2912/ DO - 10.5802/aif.2912 LA - en ID - AIF_2014__64_6_2265_0 ER -
%0 Journal Article %A Le Donne, Enrico %A Ottazzi, Alessandro %A Warhurst, Ben %T Ultrarigid tangents of sub-Riemannian nilpotent groups %J Annales de l'Institut Fourier %D 2014 %P 2265-2282 %V 64 %N 6 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2912/ %R 10.5802/aif.2912 %G en %F 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) no. 6, pp. 2265-2282. doi : 10.5802/aif.2912. http://www.numdam.org/articles/10.5802/aif.2912/
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