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

Abstract

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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.

How to cite

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Le 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

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  1. Luca Capogna, Michael Cowling, Conformality and Q -harmonicity in Carnot groups, Duke Math. J. 135 (2006), 455-479 Zbl1106.30011MR2272973
  2. 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
  3. 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
  4. John Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), 35-45 Zbl0554.53023MR806700
  5. Alessandro Ottazzi, Ben Warhurst, Contact and 1-quasiconformal maps on Carnot groups, J. Lie Theory 21 (2011), 787-811 Zbl1254.22006MR2917692
  6. Pierre Pansu, Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems 3 (1983), 415-445 Zbl0509.53040MR741395
  7. Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), 1-60 Zbl0678.53042MR979599
  8. Yehuda Shalom, Harmonic analysis, cohomology, and the large-scale geometry of amenable groups, Acta Math. 192 (2004), 119-185 Zbl1064.43004MR2096453
  9. Noboru Tanaka, On differential systems, graded Lie algebras and pseudogroups, J. Math. Kyoto Univ. 10 (1970), 1-82 Zbl0206.50503MR266258
  10. A. N. Varčenko, Obstructions to local equivalence of distributions, Mat. Zametki 29 (1981), 939-947, 957 Zbl0471.58004MR625098
  11. Ben Warhurst, Contact and Pansu differentiable maps on Carnot groups, Bull. Aust. Math. Soc. 77 (2008), 495-507 Zbl1152.22008MR2454980
  12. 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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