### Abstract

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 language | English (US) |
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Pages (from-to) | 15-39 |

Number of pages | 25 |

Journal | Analysis and Geometry in Metric Spaces |

Volume | 3 |

Issue number | 1 |

DOIs | |

State | Published - 2015 |

### Keywords

- Convergent inverse systems
- Metric measure graphs
- Pi space

### ASJC Scopus subject areas

- Analysis
- Geometry and Topology
- Applied Mathematics