Mathematical Analysis
Gromov's dimension comparison problem on Carnot groups
Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 135-138.

We solve Gromov's dimension comparison problem on Carnot groups equipped with a Carnot–Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot–Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups.

Nous présentons la solution du problème de dimension comparaison de Gromov sur les groupes de Carnot muni d'une métrique de Carnot–Carathéodory et une métrique adaptée Euclidienne. Les preuves uilisent des théorèmes de couvrir précises entre des boules Euclidienne et de Carnot–Carathéodory. Nous utilisons aussi des elements de la géométrie fractale sous-Riemanienne associée des fonctions itérées sur les groupes de Carnot.

Accepted:
Published online:
DOI: 10.1016/j.crma.2008.01.002
Balogh, Zoltán M. 1; Tyson, Jeremy T. 2; Warhurst, Ben 3

1 Department of Mathematics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
2 Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801, USA
3 School of Mathematics, University of New South Wales, Sydney 2052, Australia
@article{CRMATH_2008__346_3-4_135_0,
author = {Balogh, Zolt\'an M. and Tyson, Jeremy T. and Warhurst, Ben},
title = {Gromov's dimension comparison problem on {Carnot} groups},
journal = {Comptes Rendus. Math\'ematique},
pages = {135--138},
publisher = {Elsevier},
volume = {346},
number = {3-4},
year = {2008},
doi = {10.1016/j.crma.2008.01.002},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.01.002/}
}
TY  - JOUR
AU  - Balogh, Zoltán M.
AU  - Tyson, Jeremy T.
AU  - Warhurst, Ben
TI  - Gromov's dimension comparison problem on Carnot groups
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 135
EP  - 138
VL  - 346
IS  - 3-4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2008.01.002/
DO  - 10.1016/j.crma.2008.01.002
LA  - en
ID  - CRMATH_2008__346_3-4_135_0
ER  - 
%0 Journal Article
%A Balogh, Zoltán M.
%A Tyson, Jeremy T.
%A Warhurst, Ben
%T Gromov's dimension comparison problem on Carnot groups
%J Comptes Rendus. Mathématique
%D 2008
%P 135-138
%V 346
%N 3-4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.01.002/
%R 10.1016/j.crma.2008.01.002
%G en
%F CRMATH_2008__346_3-4_135_0
Balogh, Zoltán M.; Tyson, Jeremy T.; Warhurst, Ben. Gromov's dimension comparison problem on Carnot groups. Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 135-138. doi : 10.1016/j.crma.2008.01.002. http://www.numdam.org/articles/10.1016/j.crma.2008.01.002/

[1] Balogh, Z.M.; Hofer-Isenegger, R.; Tyson, J.T. Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group, Ergodic Theory Dynam. Systems, Volume 26 (2006), pp. 621-651

[2] Balogh, Z.M.; Rickly, M.; Serra-Cassano, F. Comparison of Hausdorff measures with respect to the Euclidean and Heisenberg metric, Publ. Mat., Volume 47 (2003), pp. 237-259

[3] Balogh, Z.M.; Tyson, J.T. Hausdorff dimensions of self-similar and self-affine fractals in the Heisenberg group, Proc. London Math. Soc. (3), Volume 91 (2005) no. 1, pp. 153-183

[4] Z.M. Balogh, J.T. Tyson, B. Warhurst, Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups, preprint, August 2007

[5] Falconer, K.J. The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc., Volume 103 (1988) no. 2, pp. 339-350

[6] Gromov, M. Carnot–Carathéodory spaces seen from within, Sub-Riemannian Geometry, Progress in Mathematics, vol. 144, Birkhäuser, Basel, 1996, pp. 79-323

[7] Strichartz, R.S. Self-similarity on nilpotent Lie groups, Philadelphia, PA, 1991 (Contemp. Math.), Volume vol. 140, Amer. Math. Soc., Providence, RI (1992), pp. 123-157

[8] Warhurst, B. Jet spaces as nonrigid Carnot groups, J. Lie Theory, Volume 15 (2005) no. 1, pp. 341-356

Cited by Sources: