Invertible Carnot Groups

David M. Freeman

Analysis and Geometry in Metric Spaces (2014)

  • Volume: 2, Issue: 1, page 248-257, electronic only
  • ISSN: 2299-3274

Abstract

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We characterize Carnot groups admitting a 1-quasiconformal metric inversion as the Lie groups of Heisenberg type whose Lie algebras satisfy the J2-condition, thus characterizing a special case of inversion invariant bi-Lipschitz homogeneity. A more general characterization of inversion invariant bi-Lipschitz homogeneity for certain non-fractal metric spaces is also provided.

How to cite

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David M. Freeman. "Invertible Carnot Groups." Analysis and Geometry in Metric Spaces 2.1 (2014): 248-257, electronic only. <http://eudml.org/doc/267299>.

@article{DavidM2014,
abstract = {We characterize Carnot groups admitting a 1-quasiconformal metric inversion as the Lie groups of Heisenberg type whose Lie algebras satisfy the J2-condition, thus characterizing a special case of inversion invariant bi-Lipschitz homogeneity. A more general characterization of inversion invariant bi-Lipschitz homogeneity for certain non-fractal metric spaces is also provided.},
author = {David M. Freeman},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {metric inversion; bi-Lipschitz homogeneity; Carnot groups; sub-Riemannian geometry},
language = {eng},
number = {1},
pages = {248-257, electronic only},
title = {Invertible Carnot Groups},
url = {http://eudml.org/doc/267299},
volume = {2},
year = {2014},
}

TY - JOUR
AU - David M. Freeman
TI - Invertible Carnot Groups
JO - Analysis and Geometry in Metric Spaces
PY - 2014
VL - 2
IS - 1
SP - 248
EP - 257, electronic only
AB - We characterize Carnot groups admitting a 1-quasiconformal metric inversion as the Lie groups of Heisenberg type whose Lie algebras satisfy the J2-condition, thus characterizing a special case of inversion invariant bi-Lipschitz homogeneity. A more general characterization of inversion invariant bi-Lipschitz homogeneity for certain non-fractal metric spaces is also provided.
LA - eng
KW - metric inversion; bi-Lipschitz homogeneity; Carnot groups; sub-Riemannian geometry
UR - http://eudml.org/doc/267299
ER -

References

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  1. [1] Jürgen Berndt, Franco Tricerri, and Lieven Vanhecke. Generalized Heisenberg groups and Damek-Ricci harmonic spaces, volume 1598 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1995. Zbl0818.53067
  2. [2] Mario Bonk and Bruce Kleiner. Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math., 150(1):127-183, 2002. Zbl1037.53023
  3. [3] Mario Bonk and Bruce Kleiner. Rigidity for quasi-Möbius group actions. J. Diferential Geom., 61(1):81-106, 2002. Zbl1044.37015
  4. [4] Mario Bonk and Bruce Kleiner. Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary. Geom. Topol., 9:219-246 (electronic), 2005. Zbl1087.20033
  5. [5] Stephen M. Buckley, David A. Herron, and Xiangdong Xie. Metric space inversions, quasihyperbolic distance, and uniform spaces. Indiana Univ. Math. J., 57(2):837-890, 2008. Zbl1160.30006
  6. [6] Sergei Buyalo and Viktor Schroeder. Möbius characterization of the boundary at infinity of rank one symmetric spaces. Geom. Dedicata, pages 1-45, 2013. Zbl06357997
  7. [7] Luca Capogna and Michael Cowling. Conformality and Q-harmonicity in Carnot groups. DukeMath. J., 135(3):455-479, 2006. Zbl1106.30011
  8. [8] Michael Cowling, Anthony H. Dooley, Adam Korányi, and Fulvio Ricci. H-type groups and Iwasawa decompositions. Adv. Math., 87(1):1-41, 1991. Zbl0761.22010
  9. [9] Michael Cowling, Anthony H. Dooley, Adam Korányi, and Fulvio Ricci. An approach to symmetric spaces of rank one via groups of Heisenberg type. J. Geom. Anal., 8(2):199-237, 1998.[Crossref] Zbl0966.53039
  10. [10] Michael Cowling and Alessandro Ottazzi. Conformal maps of Carnot groups. preprint, page arXiv:1312.6423, 2013. 
  11. [11] David M. Freeman. Inversion invariant bilipschitz homogeneity. Michigan Math. J., 61(2):415-430, 2012. Zbl1264.30046
  12. [12] David M. Freeman. Transitive bi-Lipschitz group actions and bi-Lipschitz parameterizations. Indiana Univ. Math. J., 62(1):311-331, 2013.[WoS] Zbl1294.30105
  13. [13] Juha Heinonen and Pekka Koskela. Quasiconformalmaps in metric spaceswith controlled geometry. ActaMath., 181(1):1-61, 1998. Zbl0915.30018
  14. [14] Kyle Edward Kinneberg. Rigidity for quasi-Möbius actions on fractal metric spaces. arXiv:1308:0639, 2013. Zbl1328.53052
  15. [15] Linus Kramer. Two-transitive Lie groups. J. Reine Angew. Math., 563:83-113, 2003. Zbl1044.22014
  16. [16] Enrico Le Donne. Geodesic manifolds with a transitive subset of smooth biLipschitz maps. Groups Geom. Dyn., 5(3):567-602, 2011. Zbl1242.53034
  17. [17] Enrico Le Donne. Metric spaces with unique tangents. Ann. Acad. Sci. Fenn. Math., 36(2):683-694, 2011.[Crossref] Zbl1242.54013
  18. [18] Jussi Väisälä. Quasi-Möbius maps. J. Analyse Math., 44:218-234, 1984/85. 
  19. [19] Xiangdong Xie. Quasiconformal maps on model Filiform groups. preprint, page arXiv:1308.3027, 2013. 
  20. [20] Xiangdong Xie. Quasiconformal maps on non-rigid Carnot groups. preprint, page arXiv:1308.3031, 2013. 
  21. [21] Xiangdong Xie. Rigidity of quasiconformal maps on Carnot groups. preprint, page arXiv:1308.3028, 2013. 

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