Comptes Rendus
no. 3-4
Mathematical Analysis
Gromov's dimension comparison problem on Carnot groups
[Le problème de dimension comparaison de Gromov sur les groupes de Carnot]

Présenté par : Étienne Ghys

Zoltán M. Balogh1 ; Jeremy T. Tyson2 ; Ben Warhurst3
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
Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 135-138.

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.

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.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2008.01.002
Zoltán M. Balogh 1 ; Jeremy T. Tyson 2 ; Ben Warhurst 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 
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     title = {Gromov's dimension comparison problem on {Carnot} groups},
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Zoltán M. Balogh; Jeremy T. Tyson; Ben Warhurst. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.01.002/

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

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

[3] Z.M. Balogh; J.T. Tyson 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] K.J. Falconer The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc., Volume 103 (1988) no. 2, pp. 339-350

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

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

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

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