Ultrarigid tangents of sub-Riemannian nilpotent groups
Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 2265-2282.

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.

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.

DOI: 10.5802/aif.2912
Classification: 53C17, 30L10, 22E25, 26A16
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
Le Donne, Enrico 1; Ottazzi, Alessandro 2; Warhurst, Ben 3

1 University of Jyväskylä Department of Mathematics and Statistics 40014 Jyväskylä (Finland)
2 CIRM Fondazione Bruno Kessler Via Sommarive 14 38123 Trento (Italy)
3 University of Warsaw Faculty of Mathematics Infomatics and Mechanics Banacha 2, 02-097 Warsaw (Poland)
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Le Donne, Enrico; Ottazzi, Alessandro; Warhurst, Ben. Ultrarigid tangents of sub-Riemannian nilpotent groups. Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 2265-2282. doi : 10.5802/aif.2912. https://aif.centre-mersenne.org/articles/10.5802/aif.2912/

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