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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

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IMPACT FACTOR 2018: 0.536

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Inverse Limit Spaces Satisfying a Poincaré Inequality

Jeff Cheeger / Bruce Kleiner
Published Online: 2015-01-16 | DOI: https://doi.org/10.1515/agms-2015-0002


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].

Keywords: Convergent inverse systems; metric measure graphs; PI space

MSC: 26; 28; 51


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About the article

Received: 2014-06-06

Accepted: 2014-11-25

Published Online: 2015-01-16

Citation Information: Analysis and Geometry in Metric Spaces, Volume 3, Issue 1, ISSN (Online) 2299-3274, DOI: https://doi.org/10.1515/agms-2015-0002.

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© 2015 Jeff Cheeger and Bruce Kleiner. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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