Dimensions of locally and asymptotically self-similar spaces
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S. Buyalo and N. Lebedeva
Translated by: S. Buyalo - St. Petersburg Math. J. 19 (2008), 45-65
- DOI: https://doi.org/10.1090/S1061-0022-07-00985-5
- Published electronically: December 12, 2007
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Abstract:
Two results are obtained, in a sense dual to each other. First, the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, a metric space asymptotically similar to its compact subspace has asymptotic dimension equal to the topological dimension of the subspace. As an application of the first result, the following Gromov conjecture is proved: the asymptotic dimension of every hyperbolic group $G$ equals the topological dimension of its boundary at infinity plus 1, $\operatorname {asdim} G=\dim \partial _{\infty }G+1$. As an application of the second result, we construct Pontryagin surfaces for the asymptotic dimension; in particular, these surfaces are examples of metric spaces $X$, $Y$ with $\operatorname {asdim}(X\times Y)< \operatorname {asdim}X+\operatorname {asdim}Y$. Other applications are also given.References
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Bibliographic Information
- S. Buyalo
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
- Email: sbuyalo@pdmi.ras.ru
- N. Lebedeva
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
- Email: lebed@pdmi.ras.ru
- Received by editor(s): September 29, 2005
- Published electronically: December 12, 2007
- Additional Notes: Both authors were supported by RFBR (grant 05-01-00939) and by grant NSH-1914.2003.1
- © Copyright 2007 American Mathematical Society
- Journal: St. Petersburg Math. J. 19 (2008), 45-65
- MSC (2000): Primary 51F99, 55M10
- DOI: https://doi.org/10.1090/S1061-0022-07-00985-5
- MathSciNet review: 2319510
Dedicated: To dear Viktor Abramovich Zalgaller, on the occasion of his 85th birthday