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.

Received:
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
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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/

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