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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2007, Volume 19, Issue 1, Pages 60–92 (Mi aa103)

Research Papers

Dimensions of locally and asymptotically self-similar spaces

S. V. Buyalo, N. D. Lebedeva

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

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.

Keywords: Asymptotic dimension, self-similar spaces.

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English version:
St. Petersburg Mathematical Journal, 2008, 19:1, 45–65

Bibliographic databases:

MSC: 51F99, 55M10

Citation: S. V. Buyalo, N. D. Lebedeva, “Dimensions of locally and asymptotically self-similar spaces”, Algebra i Analiz, 19:1 (2007), 60–92; St. Petersburg Math. J., 19:1 (2008), 45–65

Citation in format AMSBIB
\Bibitem{BuyLeb07} \by S.~V.~Buyalo, N.~D.~Lebedeva \paper Dimensions of locally and asymptotically self-similar spaces \jour Algebra i Analiz \yr 2007 \vol 19 \issue 1 \pages 60--92 \mathnet{http://mi.mathnet.ru/aa103} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2319510} \zmath{https://zbmath.org/?q=an:1145.54029} \transl \jour St. Petersburg Math. J. \yr 2008 \vol 19 \issue 1 \pages 45--65 \crossref{https://doi.org/10.1090/S1061-0022-07-00985-5} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000267653000004} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
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6. Mackay J.M. Sisto A., “Embedding Relatively Hyperbolic Groups in Products of Trees”, Algebr. Geom. Topol., 13:4 (2013), 2261–2282
7. Sawicki D., “Remarks on Coarse Triviality of Asymptotic Assouad-Nagata Dimension”, Topology Appl., 167 (2014), 69–75
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9. Osajda D., Swiatkowski J., “on Asymptotically Hereditarily Aspherical Groups”, Proc. London Math. Soc., 111:1 (2015), 93–126
10. Guilbault C.R., Moran M.A., “a Comparison of Large Scale Dimension of a Metric Space To the Dimension of Its Boundary”, Topology Appl., 199 (2016), 17–22
11. Dydak J., Virk Z., “Inducing Maps Between Gromov Boundaries”, Mediterr. J. Math., 13:5 (2016), 2733–2752
12. Moran M.A., “Metrics on visual boundaries of CAT(0) spaces”, Geod. Dedic., 183:1 (2016), 123–142
13. Cordes M. Hume D., “Stability and the Morse Boundary”, J. Lond. Math. Soc.-Second Ser., 95:3 (2017), 963–988
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