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Comptes Rendus Mathématique
Volume 346, n° 3-4
pages 135-138 (février 2008)
Doi : 10.1016/j.crma.2008.01.002
Received : 6 December 2006 ;  accepted : 7 January 2008
Gromov’s dimension comparison problem on Carnot groups
Le problème de dimension comparaison de Gromov sur les groupes de Carnot
 

Zoltán M. Balogh a, 1 , Jeremy T. Tyson b, 2 , Ben Warhurst c, 3
a Department of Mathematics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland 
b Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801, USA 
c School of Mathematics, University of New South Wales, Sydney 2052, Australia 

Abstract

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. To cite this article: Z.M. Balogh et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.
Résumé

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. Pour citer cet article : Z.M. Balogh et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.
1  Z.M. Balogh supported by the Swiss Nationalfond, European Research Council project GALA, and European Science Foundation project HCAA.
2  J.T. Tyson supported by NSF grant DMS 0555869.
3  B. Warhurst supported by ARC Discovery grant “Geometry of Nilpotent Groups”.


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