Inverse limit spaces satisfying a poincaré inequality

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We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [12]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces, and a large class of new examples. As follows easily from [4], generically our examples have the property that they do not bilipschitz embed in any Banach space with Radon-Nikodym property. For Laakso spaces, thiswas noted in [4]. However according to [7] these spaces admit a bilipschitz embedding in L1. For Laakso spaces, this was announced in [5].

Original languageEnglish (US)
Pages (from-to)15-39
Number of pages25
JournalAnalysis and Geometry in Metric Spaces
Issue number1
StatePublished - 2015


  • Convergent inverse systems
  • Metric measure graphs
  • Pi space

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology
  • Applied Mathematics


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