  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.
