Inverse limit spaces satisfying a Poincare inequality
Abstract
We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs (and certain higher dimensional inverse systems of metric measure spaces) 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 Poincare inequality in the sense of Heinonen-Koskela. 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. Generically our graph examples have the property that they do not bilipschitz embed in any Banach space with Radon-Nikodym property, but they do embed in the Banach space L_1. For Laakso spaces, these facts were discussed in our earlier papers.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2013
- DOI:
- 10.48550/arXiv.1312.5227
- arXiv:
- arXiv:1312.5227
- Bibcode:
- 2013arXiv1312.5227C
- Keywords:
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- Mathematics - Metric Geometry