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

This article is cited in 13 scientific papers (total in 13 papers)

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
Received: 29.09.2005

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
\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
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 1
\pages 45--65

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    This publication is cited in the following articles:
    1. N. Lebedeva, “Dimensions of products of hyperbolic spaces”, St. Petersburg Math. J., 19:1 (2008), 107–124  mathnet  crossref  mathscinet  zmath  isi
    2. Dranishnikov A., “On asymptotic dimension of amalgamated products and right-angled Coxeter groups”, Algebr. Geom. Topol., 8:3 (2008), 1281–1293  crossref  mathscinet  zmath  isi  elib
    3. Dymara J., Schick T., “Buildings have finite asymptotic dimension”, Russ. J. Math. Phys., 16:3 (2009), 409–412  crossref  mathscinet  zmath  isi
    4. Wright N., “Finite asymptotic dimension for CAT(0) cube complexes”, Geom. Topol., 16:1 (2012), 527–554  crossref  mathscinet  zmath  isi  elib
    5. Higes J., Peng I., “Assouad–Nagata dimension of connected Lie groups”, Math. Z., 273:1-2 (2013), 283–302  crossref  mathscinet  zmath  isi  elib
    6. Mackay J.M. Sisto A., “Embedding Relatively Hyperbolic Groups in Products of Trees”, Algebr. Geom. Topol., 13:4 (2013), 2261–2282  crossref  mathscinet  zmath  isi  elib
    7. Sawicki D., “Remarks on Coarse Triviality of Asymptotic Assouad-Nagata Dimension”, Topology Appl., 167 (2014), 69–75  crossref  mathscinet  zmath  isi  elib
    8. Webb R.C.H., “Combinatorics of Tight Geodesics and Stable Lengths”, Trans. Am. Math. Soc., 367:10 (2015), 7323–7342  crossref  mathscinet  zmath  isi
    9. Osajda D., Swiatkowski J., “on Asymptotically Hereditarily Aspherical Groups”, Proc. London Math. Soc., 111:1 (2015), 93–126  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    11. Dydak J., Virk Z., “Inducing Maps Between Gromov Boundaries”, Mediterr. J. Math., 13:5 (2016), 2733–2752  crossref  mathscinet  zmath  isi  scopus
    12. Moran M.A., “Metrics on visual boundaries of CAT(0) spaces”, Geod. Dedic., 183:1 (2016), 123–142  crossref  mathscinet  zmath  isi  scopus
    13. Cordes M. Hume D., “Stability and the Morse Boundary”, J. Lond. Math. Soc.-Second Ser., 95:3 (2017), 963–988  crossref  mathscinet  zmath  isi  scopus
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