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.
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Zoltán M. Balogh 1; Jeremy T. Tyson 2; Ben Warhurst 3
@article{CRMATH_2008__346_3-4_135_0, author = {Zolt\'an M. Balogh and Jeremy T. Tyson and Ben Warhurst}, 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}, }
TY - JOUR AU - Zoltán M. Balogh AU - Jeremy T. Tyson AU - Ben Warhurst 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 DO - 10.1016/j.crma.2008.01.002 LA - en ID - CRMATH_2008__346_3-4_135_0 ER -
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/
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