### 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) |
---|---|

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 |

### Fingerprint

### Keywords

- Convergent inverse systems
- Metric measure graphs
- Pi space

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Geometry and Topology

### Cite this

**Inverse limit spaces satisfying a poincaré inequality.** / Cheeger, Jeff; Kleiner, Bruce.

Research output: Contribution to journal › Article

*Analysis and Geometry in Metric Spaces*, vol. 3, no. 1, pp. 15-39. https://doi.org/10.2478/agms-2015-0002

}

TY - JOUR

T1 - Inverse limit spaces satisfying a poincaré inequality

AU - Cheeger, Jeff

AU - Kleiner, Bruce

PY - 2015

Y1 - 2015

N2 - 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].

AB - 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].

KW - Convergent inverse systems

KW - Metric measure graphs

KW - Pi space

UR - http://www.scopus.com/inward/record.url?scp=85017126004&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017126004&partnerID=8YFLogxK

U2 - 10.2478/agms-2015-0002

DO - 10.2478/agms-2015-0002

M3 - Article

VL - 3

SP - 15

EP - 39

JO - Analysis and Geometry in Metric Spaces

JF - Analysis and Geometry in Metric Spaces

SN - 2299-3274

IS - 1

ER -