Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups

Zoltán M. Balogh, Jeremy T. Tyson, Ben Warhurst

Research output: Contribution to journalArticlepeer-review


We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension 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. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples.

Original languageEnglish (US)
Pages (from-to)560-619
Number of pages60
JournalAdvances in Mathematics
Issue number2
StatePublished - Jan 30 2009


  • Carnot group
  • Hausdorff dimension
  • Iterated function system
  • Self-similar fractal

ASJC Scopus subject areas

  • General Mathematics


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