# Ultrarigid tangents of sub-Riemannian nilpotent groups

Enrico Le Donne^{[1]}; Alessandro Ottazzi^{[2]}; Ben Warhurst^{[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)

Annales de l’institut Fourier (2014)

- Volume: 64, Issue: 6, page 2265-2282
- ISSN: 0373-0956

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topLe Donne, Enrico, Ottazzi, Alessandro, and Warhurst, Ben. "Ultrarigid tangents of sub-Riemannian nilpotent groups." Annales de l’institut Fourier 64.6 (2014): 2265-2282. <http://eudml.org/doc/275527>.

@article{LeDonne2014,

abstract = {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.},

affiliation = {University of Jyväskylä Department of Mathematics and Statistics 40014 Jyväskylä (Finland); CIRM Fondazione Bruno Kessler Via Sommarive 14 38123 Trento (Italy); University of Warsaw Faculty of Mathematics Infomatics and Mechanics Banacha 2, 02-097 Warsaw (Poland)},

author = {Le Donne, Enrico, Ottazzi, Alessandro, Warhurst, Ben},

journal = {Annales de l’institut Fourier},

keywords = {Sub-Riemannian geometry; metric tangents; Gromov-Hausdorff convergence; nilpotent groups; Carnot groups; quasiconformal maps; sub-Riemannian geometry},

language = {eng},

number = {6},

pages = {2265-2282},

publisher = {Association des Annales de l’institut Fourier},

title = {Ultrarigid tangents of sub-Riemannian nilpotent groups},

url = {http://eudml.org/doc/275527},

volume = {64},

year = {2014},

}

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

PB - Association des Annales de l’institut Fourier

VL - 64

IS - 6

SP - 2265

EP - 2282

AB - 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.

LA - eng

KW - Sub-Riemannian geometry; metric tangents; Gromov-Hausdorff convergence; nilpotent groups; Carnot groups; quasiconformal maps; sub-Riemannian geometry

UR - http://eudml.org/doc/275527

ER -

## References

top- Luca Capogna, Michael Cowling, Conformality and $Q$-harmonicity in Carnot groups, Duke Math. J. 135 (2006), 455-479 Zbl1106.30011MR2272973
- G. A. Margulis, G. D. Mostow, The differential of a quasi-conformal mapping of a Carnot-Carathéodory space, Geom. Funct. Anal. 5 (1995), 402-433 Zbl0910.30020MR1334873
- G. A. Margulis, G. D. Mostow, Some remarks on the definition of tangent cones in a Carnot-Carathéodory space, J. Anal. Math. 80 (2000), 299-317 Zbl0971.58004MR1771529
- John Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), 35-45 Zbl0554.53023MR806700
- Alessandro Ottazzi, Ben Warhurst, Contact and 1-quasiconformal maps on Carnot groups, J. Lie Theory 21 (2011), 787-811 Zbl1254.22006MR2917692
- Pierre Pansu, Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems 3 (1983), 415-445 Zbl0509.53040MR741395
- Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), 1-60 Zbl0678.53042MR979599
- Yehuda Shalom, Harmonic analysis, cohomology, and the large-scale geometry of amenable groups, Acta Math. 192 (2004), 119-185 Zbl1064.43004MR2096453
- Noboru Tanaka, On differential systems, graded Lie algebras and pseudogroups, J. Math. Kyoto Univ. 10 (1970), 1-82 Zbl0206.50503MR266258
- A. N. Varčenko, Obstructions to local equivalence of distributions, Mat. Zametki 29 (1981), 939-947, 957 Zbl0471.58004MR625098
- Ben Warhurst, Contact and Pansu differentiable maps on Carnot groups, Bull. Aust. Math. Soc. 77 (2008), 495-507 Zbl1152.22008MR2454980
- Keizo Yamaguchi, Differential systems associated with simple graded Lie algebras, Progress in differential geometry 22 (1993), 413-494, Math. Soc. Japan, Tokyo Zbl0812.17018MR1274961

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