no. 6
Ultrarigid tangents of sub-Riemannian nilpotent groups
[Tangents ultra-rigides des groupes nilpotents sous-riemanniens]
Le Donne, Enrico1 ; Ottazzi, Alessandro2 ; Warhurst, Ben3
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)
Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2265-2282.

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.

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) 
@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},
     mrnumber = {3331166},
     zbl = {06387339},
     language = {en},
     url = {https://aif.centre-mersenne.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  - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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. https://aif.centre-mersenne.org/articles/10.5802/aif.2912/

[1] Capogna, Luca; Cowling, Michael Conformality and Q-harmonicity in Carnot groups, Duke Math. J., Volume 135 (2006) no. 3, pp. 455-479 | DOI | MR | Zbl

[2] Margulis, G. A.; Mostow, G. D. The differential of a quasi-conformal mapping of a Carnot-Carathéodory space, Geom. Funct. Anal., Volume 5 (1995) no. 2, pp. 402-433 | DOI | MR | Zbl

[3] Margulis, G. A.; Mostow, G. D. Some remarks on the definition of tangent cones in a Carnot-Carathéodory space, J. Anal. Math., Volume 80 (2000), pp. 299-317 | DOI | MR | Zbl

[4] Mitchell, John On Carnot-Carathéodory metrics, J. Differential Geom., Volume 21 (1985) no. 1, pp. 35-45 http://projecteuclid.org/euclid.jdg/1214439462 | MR | Zbl

[5] Ottazzi, Alessandro; Warhurst, Ben Contact and 1-quasiconformal maps on Carnot groups, J. Lie Theory, Volume 21 (2011) no. 4, pp. 787-811 | MR | Zbl

[6] Pansu, Pierre Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems, Volume 3 (1983) no. 3, pp. 415-445 | DOI | MR | Zbl

[7] Pansu, Pierre Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2), Volume 129 (1989) no. 1, pp. 1-60 | DOI | MR | Zbl

[8] Shalom, Yehuda Harmonic analysis, cohomology, and the large-scale geometry of amenable groups, Acta Math., Volume 192 (2004) no. 2, pp. 119-185 | DOI | MR | Zbl

[9] Tanaka, Noboru On differential systems, graded Lie algebras and pseudogroups, J. Math. Kyoto Univ., Volume 10 (1970), pp. 1-82 | MR | Zbl

[10] Varčenko, A. N. Obstructions to local equivalence of distributions, Mat. Zametki, Volume 29 (1981) no. 6, p. 939-947, 957 | MR | Zbl

[11] Warhurst, Ben Contact and Pansu differentiable maps on Carnot groups, Bull. Aust. Math. Soc., Volume 77 (2008) no. 3, pp. 495-507 | DOI | MR | Zbl

[12] Yamaguchi, Keizo Differential systems associated with simple graded Lie algebras, Progress in differential geometry (Adv. Stud. Pure Math.), Volume 22, Math. Soc. Japan, Tokyo, 1993, pp. 413-494 | MR | Zbl

Cité par Sources :