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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Dimensions of locally and asymptotically self-similar spaces


Authors: S. Buyalo and N. Lebedeva
Translated by: S. Buyalo
Original publication: Algebra i Analiz, tom 19 (2007), nomer 1.
Journal: St. Petersburg Math. J. 19 (2008), 45-65
MSC (2000): Primary 51F99, 55M10
Published electronically: December 12, 2007
MathSciNet review: 2319510
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Abstract | References | Similar Articles | Additional Information

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.


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Additional 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

DOI: http://dx.doi.org/10.1090/S1061-0022-07-00985-5
PII: S 1061-0022(07)00985-5
Keywords: Asymptotic dimension, self-similar spaces
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
Dedicated: To dear Viktor Abramovich Zalgaller, on the occasion of his 85th birthday
Article copyright: © Copyright 2007 American Mathematical Society